Remainder of division by 45. Division of integers with remainder, rules, examples
In this article we will analyze division of integers with remainder... Let's start with the general principle of dividing integers with remainder, formulate and prove the theorem on the divisibility of integers with remainder, trace the connections between the dividend, divisor, partial 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 Remaining Integer Division
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 equal to zero (see the theory section on division of zero by an integer), and the remainder will also be equal to 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 debt is paid. 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, and.
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.