Problem 49 - College Algebra

Question

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

Step-by-Step Solution

Answer

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 1: 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.

Step 2: 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 \(17 = 5(3) + 2\).

Step 3: 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.