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Chapter 2: Problem 58

In your own words, state the Division Algorithm.

Short Answer

Expert verified
The Division Algorithm states that if we have an integer a and a positive integer b, we can find unique integers q (quotient) and r (remainder) such that a = bq + r, with 0 鈮� r < b. For example, if a = 17 and b = 5, q = 3 and r = 2 because 17 = 3*5 + 2, where 2 is in the range [0, 5).

Step by step solution

01

Define the Division Algorithm

The Division Algorithm is a theorem in mathematics that, given any integer \(a\) and a positive integer \(b\), there exist unique integers \(q\) (quotient) and \(r\) (remainder) such that a can be expressed as \(a = bq + r\), and \(0 \leq r < b\).
02

Breakdown of Division Algorithm components

In this statement, \(a\) is the number being divided, \(b\) is the number by which \(a\) is divided, \(q\) represents the quotient of the division and \(r\) represents the remainder of the division. The conditions \(0 \leq r < b\) mean that the remainder is a non-negative integer but strictly less than the divisor \(b\).
03

Illustrate with an example

To illustrate, if \(a = 17\) and \(b = 5\), then the unique integers would be \(q = 3\) and \(r = 2\), because \(17 = 5 * 3 + 2\), and \(2\) is greater than or equal to \(0\) but less than \(5\).

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Most popular questions from this chapter

Use inspection to describe each inequality's solution set. Do not solve any of the inequalities. $$(x-2)^{2}>0$$

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