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Chapter 3: Problem 59

How can the Division Algorithm be used to check the quotient and remainder in a long division problem?

Short Answer

Expert verified
The Division Algorithm allows us to verify a long division problem by plugging the divisor, quotient, and remainder into the equation \(a = bq + r\). If it holds and the left-hand side equals the dividend, then the quotient and remainder are correct.

Step by step solution

01

Understanding Division Algorithm

The Division Algorithm is a theorem in number theory used for dividing integers. It states that for any integer \(a\) and any positive integer \(b\), there exist unique integers \(q\) and \(r\) such that \(a = bq + r\) and \(0 \leq r < b\). Here, \(a\) is the dividend, \(b\) is the divisor, \(q\) is the quotient, and \(r\) is the remainder.
02

Performing Long Division

Suppose we have a long division problem where \(a\) (dividend) is 17 and \(b\) (divisor) is 4. Performing long division, we find that the quotient (\(q\)) is 4 and the remainder (\(r\)) is 1.
03

Using Division Algorithm To Check

We apply the formula of division algorithm, \(a = bq + r\). Substituting the obtained values, we get \(17 = 4 * 4 + 1\). Simplifying this, indeed gives us \(17 = 17\), which verifies our long division calculation.

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