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Chapter 3: Problem 49

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 can be used to check the quotient and remainder in a long division problem by using its formula \(a = bq + r\), where a is the dividend, b is the divisor, q is the quotient, and r is the remainder. Once the division operation is done, the values of a, b, q, and r can be substituted in the formula to verify the results. For instance, for 17 divided by 5, the algorithm can be applied as \(17 = 5(3) + 2\), which holds true thus verifying that 3 and 2 are indeed the quotient and the remainder respectively.

Step by step solution

01

Understanding the Definition of the Division Algorithm

The first step is to understand how the Division Algorithm works. The Division Algorithm is a theorem in number theory used for division. The theorem states that if a and b are any integers with b > 0, then there exist unique integers quotient q and remainder r such that \(a = bq + r\) and \(0 \leq r < b\). Where a is the dividend, b is the divisor, q is the quotient, and r is the remainder.
02

Applying the Division Algorithm to a Long Division Problem

Let's say we have a long division problem, for instance, dividing 17 by 5, yielding a quotient of 3 and a remainder of 2. By replacing the dividend a with 17, the divisor b with 5, the quotient q with 3, and the reminder r with 2 in the Division algorithm formula, we get \(17 = 5(3) + 2\).
03

Verifying the Quotient and Remainder

Calculate the right side of the equation, \(5(3) + 2 = 15 + 2 = 17\). Since both sides of the equation equal, this verifies that the quotient and remainder of 17 divided by 5 are indeed 3 and 2 respectively.

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