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Chapter 3: Problem 58

In your own words, state the Division Algorithm.

Short Answer

Expert verified
The Division Algorithm is a theorem in number theory which states that given any integer and any positive divisor, there are unique figures called the quotient and the remainder that help express the division of these numbers in the form \(a = b*q + r\), where \(a\) is the dividend, \(b\) is the divisor, \(q\) is the quotient and \(r\) is the remainder satisfying \(0 <= r < b\).

Step by step solution

01

Understanding the Division Algorithm Theorem

Firstly, it's needed to understand what Division Algorithm theorem states. It expresses that for any integer \(a\) and any positive integer \(b\), there exist unique integers \(q\) and \(r\) such that \(a= b*q + r\) where \(0 <= r < b\). Here, \(a\) is the number to be divided, \(b\) is the divisor, \(q\) is the quotient and \(r\) the remainder. This is what is known as the Division Algorithm in number theory.
02

Understanding Each Component

The integers involved in the process are all unique and possess a specific role. The divisor \(b\) is always a positive integer. The dividend \(a\) can be any integer, it can be negative, positive, or even zero. The quotient \(q\) can also be any integer. The remainder \(r\) comes out to be a non-negative integer ranging from \(0\) up to \(b-1\).
03

Expressing in Own Words

Now express the Division Algorithm in your own words. It could be something like 'The Division Algorithm is a theorem in number theory which states that given any integer and any positive divisor, there are unique figures called the quotient and the remainder that can be used to express the division of these numbers in the form \(a = b*q + r\), where \(a\) is the dividend, \(b\) is the divisor, \(q\) is the quotient and \(r\) is the remainder satisfying \(0 <= r < b\).'

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