Remainder after division by 45. Division with remainder
Let's look at a simple example:
15:5=3
In this example, the natural number 15 is divided entirelyby 3, no remainder.
Sometimes a natural number cannot be completely divided. For example, consider a task:
There were 16 toys in the closet. There were five children in the group. Each child took the same number of toys. How many toys does each child have?
Decision:
Divide the number 16 by 5 with a column and get:
We know that 16 by 5 is not divisible. The closest smaller number that is divisible by 5 is 15 and 1 in the remainder. We can write the number 15 as 5⋅3. As a result (16 - dividend, 5 - divisor, 3 - incomplete quotient, 1 - remainder). Got formula division with remainder,by which you can make verifying the decision.
a=
b⋅
c+
d
a - dividend,
b - divisor,
c - incomplete quotient,
d Is the remainder.
Answer: each child will take 3 toys and one toy will remain.
Remainder of the division
The remainder must always be less than the divisor.
If the remainder is zero when dividing, then this means that the dividend is to be divided entirely or no remainder per divisor.
If, when dividing, the remainder is greater than the divisor, this means that the found number is not the largest. There is a larger number that will divide the dividend and the remainder will be less than the divisor.
Questions on the topic "Division with remainder":
Can the remainder be greater than the divisor?
The answer is no.
Can the remainder be equal to the divisor?
The answer is no.
How to find the dividend by incomplete quotient, divisor and remainder?
Answer: the values \u200b\u200bof the incomplete quotient, divisor and remainder are substituted into the formula and we find the dividend. Formula:
a \u003d b⋅c + d
Example # 1:
Perform division with remainder and check: a) 258: 7 b) 1873: 8
Decision:
a) We divide by a column: 
258 - dividend,
7 - divisor,
36 - incomplete quotient,
6 is the remainder. Remainder less than divisor 6<7.
7⋅36+6=252+6=258
b) We divide by a column: 
1873 - dividend,
8 - divisor,
234 - incomplete quotient,
1 is the remainder. Remainder less than divisor 1<8.
Let's substitute in the formula and check if we solved the example correctly:
8⋅234+1=1872+1=1873
Example # 2:
What are the remainders obtained by dividing natural numbers: a) 3 b) 8?
Answer:
a) The remainder is less than the divisor, therefore, less than 3. In our case, the remainder can be 0, 1 or 2.
b) The remainder is less than the divisor, therefore, less than 8. In our case, the remainder can be 0, 1, 2, 3, 4, 5, 6 or 7.
Example # 3:
What is the largest remainder that can be obtained when dividing natural numbers: a) 9 b) 15?
Answer:
a) The remainder is less than the divisor, therefore, less than 9. But we need to indicate the largest remainder. That is, the closest number to the divisor. This number is 8.
b) The remainder is less than the divisor, therefore, less than 15. But we need to indicate the largest remainder. That is, the closest number to the divisor. This number is 14.
Example # 4:
Find the dividend: a) a: 6 \u003d 3 (rest 4) b) c: 24 \u003d 4 (rest 11)
Decision:
a) Let's solve using the formula:
a \u003d b⋅c + d
(a - dividend, b - divisor, c - incomplete quotient, d - remainder.)
a: 6 \u003d 3 (rest 4)
(a - dividend, 6 - divisor, 3 - incomplete quotient, 4 - remainder.) Substitute the numbers in the formula:
a \u003d 6⋅3 + 4 \u003d 22
Answer: a \u003d 22
b) Let's solve using the formula:
a \u003d b⋅c + d
(a - dividend, b - divisor, c - incomplete quotient, d - remainder.)
from: 24 \u003d 4 (rest 11)
(c - dividend, 24 - divisor, 4 - incomplete quotient, 11 - remainder.) Substitute the numbers in the formula:
c \u003d 24⋅4 + 11 \u003d 107
Answer: c \u003d 107
A task:
Wire 4m. need to be cut into pieces of 13cm. How many of these pieces will you get?
Decision:
First, you need to convert meters to centimeters.
4m. \u003d 400cm.
You can divide it by a column or in your mind we get:
400: 13 \u003d 30 (rest 10)
Let's check:
13⋅30+10=390+10=400
Answer: 30 pieces will turn out and 10 cm of wire will remain.
In this article we will analyze division of integers with remainder... Let's start with the general principle of division of integers with remainder, formulate and prove the theorem on the divisibility of integers with remainder, trace the connections between the dividend, divisor, incomplete quotient and remainder. Next, we will voice the rules by which the division of integers with a remainder is carried out, and consider the application of these rules when solving examples. After that, we will learn how to check the result of dividing integers with remainder.
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Understanding the division of integers with remainder
We will consider division of integers with remainder as a generalization of division with remainder of natural numbers. This is due to the fact that natural numbers are an integral part of integers.
Let's start with the terms and designations that are used in the description.
By analogy with dividing natural numbers with a remainder, we will assume that the result of division with a remainder of two integers a and b (b is not equal to zero) are two integers c and d. The numbers a and b are called divisible and divider respectively, the number d - the remainder from dividing a by b, and the integer c is called incomplete private (or simply privateif the remainder is zero).
Let us agree to assume that the remainder is a non-negative integer, and its value does not exceed b, that is, (we met such chains of inequalities when we talked about comparing three or more integers).
If the number c is an incomplete quotient, and the number d is the remainder of dividing an integer a by an integer b, then we will briefly write this fact as an equality of the form a: b \u003d c (remainder d).
Note that when dividing an integer a by an integer b, the remainder can be zero. In this case a is said to be divisible by b without residue (or entirely). Thus, dividing integers without a remainder is a special case of dividing integers with a remainder.
It is also worth saying that when dividing zero by some integer, we always deal with division without a remainder, since in this case the quotient will be zero (see the section on the theory of division of zero by an integer), and the remainder will also be zero.
We have decided on the terminology and designations, now let's figure out the meaning of dividing integers with a remainder.
Dividing a negative integer a by a positive integer b can also make sense. To do this, consider a negative integer as debt. Let's imagine the following situation. The debt, which constitutes the items, must be paid by b people, making the same contribution. In this case, the absolute value of the incomplete quotient c will determine the amount of debt of each of these people, and the remainder d will show how many items will remain after the payment of the debt. Let's give an example. Let's say 2 people need 7 apples. If we assume that each of them owes 4 apples, then after paying the debt they will have 1 apple. This situation corresponds to the equality (−7): 2 \u003d −4 (rest 1).
We will not give any meaning to division with the remainder of an arbitrary integer a by a negative integer, but we will leave it with the right to exist.
Divisibility theorem for integers with remainder
When we talked about dividing natural numbers with remainder, we found out that dividend a, divisor b, incomplete quotient c and remainder d are related by the equality a \u003d b c + d. The integers a, b, c, and d share the same relationship. This connection is approved by the following remainder divisibility theorem.
Theorem.
Any integer a can be represented uniquely through an integer and nonzero number b in the form a \u003d b q + r, where q and r are some integers, moreover.
Evidence.
First, we prove the possibility of representing a \u003d b q + r.
If integers a and b are such that a is evenly divisible by b, then by definition there exists an integer q such that a \u003d b q. In this case, the equality a \u003d b q + r holds for r \u003d 0.
Now we will assume that b is a positive integer. Choose an integer q such that the product b q does not exceed a, and the product b (q + 1) is already greater than a. That is, we take q such that the inequalities b q It remains to prove the possibility of representing a \u003d b q + r for negative b. Since the modulus of the number b in this case is a positive number, then for there is a representation, where q 1 is some integer, and r is an integer satisfying the conditions. Then, taking q \u003d −q 1, we obtain the required representation a \u003d b q + r for negative b. We pass to the proof of uniqueness. Suppose that, in addition to the representation a \u003d bq + r, q and r are integers and, there is one more representation a \u003d bq 1 + r 1, where q 1 and r 1 are some integers, and q 1 q and. After subtracting from the left and right sides of the first equality, respectively, the left and right sides of the second equality, we obtain 0 \u003d b (q − q 1) + r − r 1, which is equivalent to the equality r − r 1 \u003d b (q 1 −q) ... Then an equality of the form From the conditions and we can conclude that. Since q and q 1 are integers and q ≠ q 1, whence we conclude that The equality a \u003d b c + d allows you to find the unknown dividend a if you know the divisor b, the incomplete quotient c, and the remainder d. Let's look at an example. Example. What is the dividend if dividing it by the integer −21 results in an incomplete quotient 5 and a remainder of 12? Decision. We need to calculate the dividend a when we know the divisor b \u003d −21, the partial quotient c \u003d 5, and the remainder d \u003d 12. Turning to the equality a \u003d b c + d, we get a \u003d (- 21) 5 + 12. Observing, first we multiply the integers −21 and 5 according to the rule of multiplying integers with different signs, after which we add integers with different signs: (−21) 5 + 12 \u003d −105 + 12 \u003d −93. Answer: −93
.
The connections between the dividend, divisor, partial quotient and remainder are also expressed by equalities of the form b \u003d (a − d): c, c \u003d (a − d): b and d \u003d a − b · c. These equalities allow calculating the divisor, partial quotient, and remainder, respectively. We often have to find the remainder of dividing an integer a by an integer b, when the dividend, divisor, and partial quotient are known, using the formula d \u003d a − b · c. To avoid further questions, let's look at an example of calculating the remainder. Example. Find the remainder of dividing the integer −19 by the integer 3 if it is known that the incomplete quotient is −7. Decision. To calculate the remainder of the division, we use a formula of the form d \u003d a − b · c. From the condition we have all the necessary data a \u003d −19, b \u003d 3, c \u003d −7. We get d \u003d a − b c \u003d −19−3 (−7) \u003d −19 - (- 21) \u003d - 19 + 21 \u003d 2 (the difference −19 - (- 21) we calculated by the rule of subtraction of a negative integer ). Answer: As we have noted more than once, positive integers are natural numbers. Therefore, division with remainder of positive integers is carried out according to all division rules with remainder of natural numbers. It is very important to be able to easily perform division with the remainder of natural numbers, since it is this that underlies not only division of positive integers, but also the basis of all division rules with the remainder of arbitrary integers. From our point of view, it is most convenient to perform long division, this method allows you to get both the incomplete quotient (or just the quotient) and the remainder. Consider an example of division with remainder of positive integers. Example. Divide 14 671 by 54 with the remainder. Decision. Let's divide these positive integers by a column: The partial quotient turned out to be 271, and the remainder is 37. Answer: 14 671: 54 \u003d 271 (rest 37). Let us formulate a rule that allows performing division with a remainder of a positive integer by a negative integer. The incomplete quotient of dividing a positive integer a by a negative integer b is the opposite of the incomplete quotient of dividing a by the modulus of b, and the remainder of dividing a by b is equal to the remainder of division by. From this rule it follows that the incomplete quotient of dividing a positive integer by a negative integer is a non-positive integer. Let's remake the announced rule into an algorithm for division with the remainder of a positive integer by a negative integer: Let's give an example of using the algorithm for dividing a positive integer by a negative integer. Example. Divide the positive integer 17 by the negative integer −5. Decision. Let's use the algorithm of division with the remainder of a positive integer by a negative integer. Dividing The opposite of 3 is −3. Thus, the required partial quotient of 17 divided by −5 is −3, and the remainder is 2. Answer: 17: (- 5) \u003d - 3 (rest 2). Example. Divide 45 to -15. Decision. The moduli of the dividend and the divisor are 45 and 15, respectively. The number 45 is divisible by 15 without a remainder, while the quotient is 3. Therefore, the positive integer 45 is divisible by the negative integer −15 without a remainder, the quotient being equal to the opposite number of 3, that is, −3. Indeed, according to the rule of dividing integers with different signs, we have. Answer: 45:(−15)=−3
.
Let us give the formulation of the division rule with the remainder of a negative integer by a positive integer. To get an incomplete quotient c from dividing a negative integer a by a positive integer b, you need to take the opposite of the incomplete quotient from dividing the moduli of the original numbers and subtract one from it, then calculate the remainder d by the formula d \u003d a − b c. From this rule of division with a remainder, it follows that the incomplete quotient of dividing a negative integer by a positive integer is a negative integer. From the sounded rule follows the division algorithm with the remainder of a negative integer a by a positive integer b: Let us analyze the solution of the example, in which we use the written division algorithm with remainder. Example. Find the incomplete quotient and the remainder of the negative integer -17 divided by the positive integer 5. Decision. The modulus of the dividend −17 is 17, and the modulus of the divisor 5 is 5. Dividing 17 by 5, we get incomplete quotient 3 and remainder 2. The opposite of 3 is −3. Subtract one from −3: −3−1 \u003d −4. So, the required incomplete quotient is −4. It remains to calculate the remainder. In our example, a \u003d −17, b \u003d 5, c \u003d −4, then d \u003d a − b c \u003d −17−5 (−4) \u003d −17 - (- 20) \u003d - 17 + 20 \u003d 3 ... Thus, the partial quotient of dividing the negative integer -17 by the positive integer 5 is -4, and the remainder is 3. Answer: (−17): 5 \u003d −4 (rest 3). Example. Divide the negative integer -1404 by the positive integer 26. Decision. The modulus of the dividend is 1 404, the modulus of the divisor is 26. Divide 1 404 by 26 with a column: Since the modulus of the dividend was divided by the modulus of the divisor without a remainder, the original integers are divisible without a remainder, and the desired quotient is equal to the number opposite to 54, that is, −54. Answer: (−1 404):26=−54
.
Let's formulate the division rule with remainder of negative integers. To get an incomplete quotient c from dividing a negative integer a by an integer negative number b, you need to calculate the incomplete quotient from dividing the moduli of the original numbers and add one to it, then calculate the remainder d by the formula d \u003d a − b c. It follows from this rule that the incomplete quotient of the division of negative integers is a positive integer. Let's rewrite the stated rule as an algorithm for dividing negative integers: Consider using the algorithm for dividing negative integers when solving an example. Example. Find the partial quotient and the remainder of the division of the negative integer -17 by the negative integer -5. Decision. Let's use the appropriate division with remainder algorithm. The modulus of the dividend is 17, the modulus of the divisor is 5. Division 17 by 5 gives an incomplete quotient 3 and a remainder of 2. Add one to the incomplete quotient 3: 3 + 1 \u003d 4. Therefore, the required incomplete quotient of dividing −17 by −5 is 4. It remains to calculate the remainder. In this example, a \u003d −17, b \u003d −5, c \u003d 4, then d \u003d a − b c \u003d −17 - (- 5) 4 \u003d −17 - (- 20) \u003d - 17 + 20 \u003d 3 ... So, the incomplete quotient of dividing the negative integer -17 by the negative integer -5 is 4, and the remainder is 3. Answer: (−17): (- 5) \u003d 4 (rest 3). After dividing the integers with remainder, it is useful to check the result. The check is carried out in two stages. At the first stage, it is checked whether the remainder d is a non-negative number, and the condition is also checked. If all the conditions of the first stage of verification are met, then you can proceed to the second stage of verification, otherwise it can be argued that a mistake was made somewhere during division with a remainder. At the second stage, the validity of the equality a \u003d b c + d is checked. If this equality is true, then the division with the remainder was carried out correctly, otherwise, a mistake was made somewhere. Let's consider solutions of examples in which the result of division of integers with remainder is checked. Example. When dividing the number −521 by −12, you got an incomplete quotient 44 and a remainder of 7, check the result. Decision. −2 for b \u003d −3, c \u003d 7, d \u003d 1. We have b c + d \u003d −3 7 + 1 \u003d −21 + 1 \u003d −20... Thus, the equality a \u003d b c + d is incorrect (in our example, a \u003d −19). Therefore, the division with the remainder was carried out incorrectly. The article discusses the concept of division of integers with remainder. Let us prove the theorem on the divisibility of integers with remainder and examine the connections between dividends and divisors, incomplete quotients and remainders. Let's consider the rules when division of integers with remainders is performed, having considered in detail with examples. At the end of the solution, let's check. Division of integers with remainder is considered as generalized division with remainder of natural numbers. This is done because natural numbers are a constituent part of integers. Division with the remainder of an arbitrary number means that the integer a is divisible by a non-zero number b. If b \u003d 0, then no remainder division is performed. As well as division of natural numbers with remainder, division of integers a and b, if b is different from zero, by c and d is performed. In this case, a and b are called the dividend and divisor, and d is the remainder of the division, c is an integer or an incomplete quotient. If we assume that the remainder is a non-negative integer, then its value is not more than the modulus of the number b. Let's write this way: 0 ≤ d ≤ b. This chain of inequalities is used when comparing 3 or more numbers. If c is an incomplete quotient, then d is the remainder of dividing an integer a by b, you can briefly fix: a: b \u003d c (remainder d). The remainder when dividing numbers a by b is possible zero, then they say that a is divisible by b completely, that is, without a remainder. Division without remainder is considered a special case of division. If we divide zero by some number, we get zero as a result. The remainder of the division will also be zero. This can be traced back to the theory of dividing zero by an integer. Now let's look at the meaning of dividing integers with remainder. It is known that positive integers are natural, then when dividing with a remainder, you get the same meaning as when dividing natural numbers with a remainder. When dividing a negative integer a by a positive integer b makes sense. Let's look at an example. Imagining a situation where we have a debt of items in the amount a, which must be repaid by b people. This requires everyone to make the same contribution. To determine the amount of debt for each, you need to pay attention to the amount of private p. The remainder d says that the number of items is known after paying off debts. Let's take an example with apples. If 2 people need 7 apples. If you count that everyone must return 4 apples, after the full calculation they will have 1 apple. Let us write it in the form of an equality: (- 7): 2 \u003d - 4 (o with point 1). Division of any number a by an integer does not make sense, but it is possible as an option. We found that a is a dividend, then b is a divisor, c is an incomplete quotient, and d is a remainder. They are related to each other. We will show this connection using the equality a \u003d b c + d. The connection between them is characterized by the remainder divisibility theorem. Theorem Any integer can only be represented through an integer and nonzero number b in this way: a \u003d b q + r, where q and r are some integers. Here we have 0 ≤ r ≤ b. Let us prove the possibility of the existence of a \u003d b q + r. Evidence If there are two numbers a and b, and a is divisible by b without a remainder, then the definition implies that there is a number q, which will be true the equality a \u003d b q. Then the equality can be considered true: a \u003d b q + r for r \u003d 0. Then it is necessary to take q such that given by the inequality b q< a < b · (q + 1) было верным. Необходимо вычесть b · q из всех частей выражения. Тогда придем к неравенству такого вида: 0 < a − b · q < b . We have that the value of the expression a - b q is greater than zero and not greater than the value of the number b, it follows that r \u003d a - b q. We get that the number a can be represented as a \u003d b q + r. It is now necessary to consider the possibility of representing a \u003d b q + r for negative values \u200b\u200bof b. The absolute value of the number is positive, then we get a \u003d b q 1 + r, where the value q 1 is some integer, r is an integer that matches the condition 0 ≤ r< b . Принимаем q = − q 1 , получим, что a = b · q + r для отрицательных b . Proof of uniqueness Suppose a \u003d bq + r, q and r are integers with the true condition 0 ≤ r< b , имеется еще одна форма записи в виде a = b · q 1 + r 1 , где q 1 and r 1 are some numbers, where q 1 ≠ q , 0 ≤ r 1< b . When the inequality is subtracted from the left and right sides, then we get 0 \u003d b · (q - q 1) + r - r 1, which is equivalent to r - r 1 \u003d b · q 1 - q. Since the modulus is used, we obtain the equality r - r 1 \u003d b q 1 - q. The given condition says that 0 ≤ r< b и 0 ≤ r 1 < b запишется в виде r - r 1 < b . Имеем, что qand q 1- integers, and q ≠ q 1, then q 1 - q ≥ 1. Hence we have that b q 1 - q ≥ b. The resulting inequalities r - r 1< b и b · q 1 - q ≥ b указывают на то, что такое равенство в виде r - r 1 = b · q 1 - q невозможно в данном случае. Hence it follows that the number a cannot be represented in any other way, except by such a notation a \u003d b q + r. Using the equality a \u003d b c + d, you can find the unknown dividend a when you know the divisor b with incomplete quotient c and remainder d. Example 1 Determine the dividend if in division we get - 21, incomplete quotient 5 and remainder 12. Decision
It is necessary to calculate the dividend a with the known divisor b \u003d - 21, incomplete quotient c \u003d 5 and remainder d \u003d 12. It is necessary to turn to the equality a \u003d b c + d, from which we get a \u003d (- 21) 5 + 12. Subject to the order of performing the actions, we multiply - 21 by 5, after that we get (- 21) 5 + 12 \u003d - 105 + 12 \u003d - 93. Answer: - 93 . The connection between the divisor and the incomplete quotient and the remainder can be expressed using the equalities: b \u003d (a - d): c, c \u003d (a - d): b and d \u003d a - b c. With their help, we can calculate the divisor, partial quotient and remainder. This boils down to constantly finding the remainder after dividing an integer a by b with a known dividend, divisor, and incomplete quotient. The formula applies d \u003d a - b c. Let's consider the solution in detail. Example 2 Find the remainder of dividing an integer - 19 by an integer 3 with a known incomplete quotient equal to - 7. Decision
To calculate the remainder of the division, apply a formula like d \u003d a - b · c. By condition, all data are available a \u003d - 19, b \u003d 3, c \u003d - 7. From this we get d \u003d a - b c \u003d - 19 - 3 (- 7) \u003d - 19 - (- 21) \u003d - 19 + 21 \u003d 2 (the difference is - 19 - (- 21). This example is calculated by the subtraction rule an integer negative number. Answer: 2 . All positive integers are natural. It follows that division is performed according to all division rules with the remainder of natural numbers. The speed of division with the remainder of natural numbers is important, since not only the division of positive ones, but also the rules for dividing arbitrary integers are based on it. The most convenient method of division is a column, since it is easier and faster to get an incomplete or just a quotient with a remainder. Let's consider the solution in more detail. Example 3 Divide 14671 by 54. Decision
This division must be performed in a column: That is, the incomplete quotient turns out to be 271, and the remainder is 37. Answer: 14 671: 54 \u003d 271. (stop 37) To divide with a positive remainder by a negative integer, you need to formulate a rule. Definition 1 Incomplete quotient from dividing a positive integer a by a negative integer b we get a number that is opposite to the incomplete quotient from dividing the absolute values \u200b\u200bof numbers a by b. Then the remainder is equal to the remainder when a is divided by b. Hence, we have that the incomplete quotient of dividing an integer positive number by an integer negative number is considered a non-positive integer. We get the algorithm: Let's consider an example of the algorithm for dividing a positive integer by a negative integer. Example 4 Divide with a remainder of 17 by - 5. Decision
Let's apply the algorithm of division with the remainder of a positive integer by a negative integer. You must divide 17 by - 5 modulo. From this we get that the incomplete quotient is 3, and the remainder is 2. We get that the required number from dividing 17 by - 5 \u003d - 3 with a remainder of 2. Answer: 17: (- 5) \u003d - 3 (rest 2). Example 5 Divide 45 by - 15. Decision
It is necessary to divide the numbers modulo. Divide the number 45 by 15, we get the quotient 3 without a remainder. This means that the number 45 is divisible by 15 without a remainder. In the answer we get - 3, since the division was performed modulo. 45: (- 15) = 45: - 15 = - 45: 15 = - 3 Answer: 45: (− 15) = − 3 . The formulation of the division rule with remainder is as follows. Definition 2 In order to get an incomplete quotient c when dividing a negative integer a by a positive b, you need to apply the opposite of the given number and subtract 1 from it, then the remainder d will be calculated by the formula: d \u003d a - b · c. Based on the rule, we can conclude that when dividing we get a non-negative integer number. For the accuracy of the solution, the algorithm for dividing a by b with a remainder is used: Let's consider an example of a solution where this algorithm is applied. Example 6 Find the incomplete quotient and the remainder of the division - 17 by 5. Decision
Divide the given numbers modulo. We get that when dividing the quotient is 3, and the remainder is 2. Since we got 3, the opposite is 3. You must subtract 1. − 3 − 1 = − 4 . We get the desired value equal to - 4. To calculate the remainder, you need a \u003d - 17, b \u003d 5, c \u003d - 4, then d \u003d a - b c \u003d - 17 - 5 (- 4) \u003d - 17 - (- 20) \u003d - 17 + 20 \u003d 3. This means that the incomplete quotient of division is the number - 4 with a remainder equal to 3. Answer: (- 17): 5 \u003d - 4 (rest. 3). Example 7 Divide negative integer 1404 by positive 26. Decision
It is necessary to divide by a column and by a mule. We got the division of the absolute values \u200b\u200bof numbers without remainder. This means that the division is performed without a remainder, and the desired quotient \u003d - 54. Answer: (− 1 404) : 26 = − 54 . It is necessary to formulate a division rule with remainder of negative integers. Definition 3 To obtain an incomplete quotient c from dividing a negative integer a by an integer negative b, it is necessary to perform calculations modulo, then add 1, then we can perform calculations using the formula d \u003d a - b · c. It follows that the incomplete quotient of dividing negative integers will be a positive number. Let's formulate this rule in the form of an algorithm: Let us consider this algorithm using an example. Example 8 Find the incomplete quotient and remainder when dividing - 17 by - 5. Decision
For the correctness of the solution, we will apply the algorithm for division with remainder. First, divide the numbers modulo. From here we get that the incomplete quotient \u003d 3, and the remainder is 2. According to the rule, it is necessary to add the incomplete quotient and 1. We get that 3 + 1 \u003d 4. From this we get that the incomplete quotient of the division of the given numbers is 4. To calculate the remainder, we will use the formula. By hypothesis, we have that a \u003d - 17, b \u003d - 5, c \u003d 4, then, using the formula, we get d \u003d a - b c \u003d - 17 - (- 5) 4 \u003d - 17 - (- 20) \u003d - 17 + 20 \u003d 3. The desired answer, that is, the remainder, is 3, and the incomplete quotient is 4. Answer: (- 17): (- 5) \u003d 4 (rest 3). After performing division of numbers with remainder, you need to check. This check involves 2 stages. First, the remainder d is checked for nonnegativity, the condition 0 ≤ d< b . При их выполнении разрешено выполнять 2 этап. Если 1 этап не выполнился, значит вычисления произведены с ошибками. Второй этап состоит из того, что равенство a = b · c + d должно быть верным. Иначе в вычисления имеется ошибка. Let's look at some examples. Example 9 The division was made - 521 by - 12. The quotient is 44, the remainder is 7. Check. Decision
Since the remainder is a positive number, its value is less than the modulus of the divisor. The divisor is - 12, which means that its modulus is 12. You can proceed to the next check point. By hypothesis, we have that a \u003d - 521, b \u003d - 12, c \u003d 44, d \u003d 7. From here we calculate b c + d, where b c + d \u003d - 12 44 + 7 \u003d - 528 + 7 \u003d - 521. Hence it follows that the equality is true. Verification passed. Example 10 Perform division check (- 17): 5 \u003d - 3 (rest - 2). Is equality true? Decision
The point of the first stage is that it is necessary to check the division of integers with remainder. From this it is clear that the action was performed incorrectly, since the remainder is given, equal to - 2. The remainder is not negative. We have that the second condition is satisfied, but insufficient for this case. Answer: not. Example 11 Number - 19 divided by - 3. The incomplete quotient is 7 and the remainder is 1. Check if the calculation is correct. Decision
A remainder of 1 is given. He's positive. It is less than the divider module, which means that the first stage is performed. Let's move on to the second stage. Let's calculate the value of the expression b c + d. By hypothesis, we have that b \u003d - 3, c \u003d 7, d \u003d 1, therefore, substituting the numerical values, we get b c + d \u003d - 3 7 + 1 \u003d - 21 + 1 \u003d - 20. It follows that a \u003d b c + d the equality does not hold, since a \u003d - 19 is given in the condition. Hence the conclusion that the division was made with an error. Answer: not. If you notice an error in the text, please select it and press Ctrl + Enter Divisibility tests for numbers- these are rules that allow you to relatively quickly find out, without division, whether this number is divisible by a given one without a remainder. This is one of the simplest divisibility tests. It sounds like this: if the recording of a natural number ends in an even digit, then it is even (divisible by 2 without a remainder), and if the recording of a number ends in an odd digit, then this number is odd. This divisibility criterion has completely different rules: if the sum of the digits of a number is divisible by 3, then the number is also divisible by 3; if the sum of the digits of a number is not divisible by 3, then the number is not divisible by 3 either. This divisibility criterion will be more complicated. If the last 2 digits of the number form a number that is divisible by 4 or it is 00, then the number is divisible by 4, otherwise this number is not divisible by 4 without a remainder. And again we have a rather simple divisibility sign: if the record of a natural number ends with a digit 0 or 5, then this number is divisible without a remainder by 5. If the record of a number ends with another digit, then the number is not divisible by 5 without a remainder. Before us is a composite number 6, which is the product of the numbers 2 and 3. Therefore, the divisibility by 6 is also composite: in order for a number to be divisible by 6, it must correspond to two divisibility features at the same time: the divisibility feature by 2 and the divisibility feature by 3. At the same time, note that such a composite number as 4 has an individual sign of divisibility, because it is the product of the number 2 by itself. But back to the divisibility by 6 criterion. This sign of divisibility is more complex: a number is divisible by 7 if the result of subtracting the last doubled digit from the tens of this number is divisible by 7 or equal to 0. Divisibility by 8 is as follows: if the last 3 digits form a number that is divisible by 8, or 000, then the given number is divisible by 8. This sign of divisibility is similar to the sign of divisibility by 3: if the sum of the digits of a number is divisible by 9, then the number is also divisible by 9; if the sum of the digits of a number is not divisible by 9, then the number is not divisible by 9 either. I combined these signs of divisibility because they can be described in the same way: a number is divided by a bit unit if the number of zeros at the end of the number is greater than or equal to the number of zeros in a given bit unit. In order to find out whether a number is divisible by 11, you need to get the difference between the sums of the even and odd digits of this number. If this difference is equal to 0 or is divisible by 11 without a remainder, then the number itself is divisible by 11 without a remainder. The number 12 is compound. Its divisibility criterion is the correspondence to the divisibility criteria by 3 and 4 simultaneously. The sign of divisibility by 13 is that if the number of tens of a number, added with multiplied by 4 units of this number, is a multiple of 13 or equal to 0, then the number itself is divisible by 13. As can be seen from the above, it can be assumed that for any of the natural numbers you can choose your own individual divisibility criterion or a "composite" feature if the number is a multiple of several different numbers. But as practice shows, in general, the larger the number, the more complex its sign. It is possible that the time spent on checking the divisibility criterion may be equal to or more than the division itself. Therefore, we usually use the simplest of divisibility criteria.
, and by virtue of the properties of the modulus of a number, the equality 
.
... From the obtained inequalities and it follows that an equality of the form impossible under our assumption. Therefore, there is no other representation of the number a, except for a \u003d b q + r.Links between dividend, divisor, incomplete quotient and remainder
Division with remainder of positive integers, examples
The rule of division with a remainder of a positive integer by a negative integer, examples
Division with remainder of a negative integer by a positive integer, examples
Division rule with remainder of negative integers, examples
Checking the result of dividing integers with remainder
Understanding Remaining Integer Division
Divisibility theorem for integers with remainder
Relationship between dividend, divisor, incomplete quotient and remainder
The rule of division with a remainder of a positive integer by a negative integer, examples
Division rule with remainder of negative integers, examples
Checking the result of dividing integers with remainder
Some of divisibility criteria quite simple, some more difficult. On this page you will find both the divisibility criteria for prime numbers, such as, for example, 2, 3, 5, 7, 11, and the divisibility criteria for composite numbers, such as 6 or 12.
I hope this information will be useful to you.
Happy learning! Divisibility by 2
In other words, if the last digit of the number is 2
, 4
, 6
, 8
or 0
- the number is divisible by 2, if not, then it is not divisible
For example, numbers: 23 4
, 8270
, 1276
, 9038
, 502
are divisible by 2 because they are even.
And numbers: 23 5
, 137
, 2303
are not divisible by 2 because they are odd. Divisibility by 3
So, in order to understand whether a number is divisible by 3, you just need to add together the numbers of which it consists.
It looks like this: 3987 and 141 are divisible by 3, because in the first case 3 + 9 + 8 + 7 \u003d 27
(27: 3 \u003d 9 - divisible by 3 without ostak), and in the second 1 + 4 + 1 \u003d 6
(6: 3 \u003d 2 - also divisible by 3 without ostak).
But the numbers: 235 and 566 are not divisible by 3, because 2 + 3 + 5 \u003d 10
and 5 + 6 + 6 \u003d 17
(and we know that neither 10 nor 17 are divisible by 3 without a remainder). Divisibility by 4
For example: 1 00
and 3 64
are divided by 4, because in the first case, the number ends in 00
, and in the second on 64
, which in turn is divisible by 4 without a remainder (64: 4 \u003d 16)
Numbers 3 57
and 8 86
are not divisible by 4, because neither 57
nor 86
are not divisible by 4, which means they do not correspond to this criterion of divisibility. Divisibility by 5
This means that any numbers ending in digits 0
and 5
e.g. 1235 5
and 43 0
, fall under the rule and are divisible by 5.
And, for example, 1549 3
and 56 4
do not end in 5 or 0, which means they cannot be divisible by 5 without a remainder. Divisibility by 6
The numbers 138 and 474 are even and correspond to the criteria of divisibility by 3 (1 + 3 + 8 \u003d 12, 12: 3 \u003d 4 and 4 + 7 + 4 \u003d 15, 15: 3 \u003d 5), which means they are divisible by 6. But 123 and 447, although they are divisible by 3 (1 + 2 + 3 \u003d 6, 6: 3 \u003d 2 and 4 + 4 + 7 \u003d 15, 15: 3 \u003d 5), but they are odd, which means they do not correspond to the divisibility criterion by 2, and therefore do not correspond to the divisibility criterion by 6. Divisibility by 7
Sounds pretty confusing, but simple in practice. See for yourself: the number 95
9 is divisible by 7 because 95
-2 * 9 \u003d 95-18 \u003d 77, 77: 7 \u003d 11 (77 is divisible by 7 without remainder). Moreover, if difficulties arose with the number obtained during the transformations (because of its size it is difficult to understand whether it is divisible by 7 or not, then this procedure can be continued as many times as you deem necessary).
For example, 45
5 and 4580
1 have signs of divisibility by 7. In the first case, everything is quite simple: 45
-2 * 5 \u003d 45-10 \u003d 35, 35: 7 \u003d 5. In the second case, we will do this: 4580
-2 * 1 \u003d 4580-2 \u003d 4578. It's hard for us to understand if 457
8 by 7, so let's repeat the process: 457
-2 * 8 \u003d 457-16 \u003d 441. And again we will use the divisibility criterion, since we still have a three-digit number 44
1. So, 44
-2 * 1 \u003d 44-2 \u003d 42, 42: 7 \u003d 6, i.e. 42 is divisible by 7 without a remainder, which means 45801 is divisible by 7.
But the numbers 11
1 and 34
5 is not divisible by 7 because 11
-2 * 1 \u003d 11 - 2 \u003d 9 (9 is not evenly divisible by 7) and 34
-2 * 5 \u003d 34-10 \u003d 24 (24 is not evenly divisible by 7). Divisibility by 8
Numbers 1 000
or 1 088
divisible by 8: the first ends in 000
, the second 88
: 8 \u003d 11 (divisible by 8 without remainder).
But the numbers 1 100
or 4 757
are not divisible by 8, since the numbers 100
and 757
are not evenly divisible by 8. Divisibility by 9
For example: 3987 and 144 are divisible by 9, because in the first case 3 + 9 + 8 + 7 \u003d 27
(27: 9 \u003d 3 - divisible by 9 without ostak), and in the second 1 + 4 + 4 \u003d 9
(9: 9 \u003d 1 - also divisible by 9 without ostak).
But the numbers: 235 and 141 are not divisible by 9, because 2 + 3 + 5 \u003d 10
and 1 + 4 + 1 \u003d 6
(and we know that neither 10 nor 6 are divisible by 9 without a remainder). Divisibility by 10, 100, 1000 and other bit units
In other words, for example, we have numbers like this: 654 0
, 46400
, 867000
, 6450
... of which all are divisible by 1 0
; 46400
and 867 000
are also divided by 1 00
; and only one of them - 867 000
divisible by 1 000
.
Any numbers that have less zeros at the end than a bit unit are not divisible by that bit unit, for example 600 30
and 7 93
not divisible 1 00
.Divisibility by 11
To make it clearer, I propose to consider examples: 2
35
4 is divisible by 11 because ( 2
+5
)-(3+4)=7-7=0. 29
19
4 is also divisible by 11, since ( 9
+9
)-(2+1+4)=18-7=11.
But 1 1
1 or 4
35
4 is not divisible by 11, since in the first case we get (1 + 1) - 1
\u003d 1, and in the second ( 4
+5
)-(3+4)=9-7=2.Divisibility by 12
For example, 300 and 636 correspond to both the signs of divisibility by 4 (the last 2 digits are zeros or are divisible by 4) and the signs of divisibility by 3 (the sum of the digits and the first and three times the number is divisible by 3), and znit, they are divisible by 12 without a remainder.
But 200 or 630 are not divisible by 12, because in the first case the number corresponds only to the divisibility by 4, and in the second - only to the divisibility by 3. but not to both signs at the same time. Divisibility by 13
Take for example 70
2. So, 70
+ 4 * 2 \u003d 78, 78: 13 \u003d 6 (78 is divisible by 13 without remainder), which means 70
2 is divisible by 13 without remainder. Another example is the number 114
4. 114
+ 4 * 4 \u003d 130, 130: 13 \u003d 10. The number 130 is divisible by 13 without a remainder, which means that the given number corresponds to the divisibility criterion by 13.
If we take the numbers 12
5 or 21
2, then we get 12
+ 4 * 5 \u003d 32 and 21
+ 4 * 2 \u003d 29, respectively, and neither 32 nor 29 are divisible by 13 without a remainder, which means that the given numbers are not evenly divisible by 13.Divisibility of numbers
