Problem 59 - 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 applied to verify the quotient and remainder in a long division problem. For instance, in the example of 17 divided by 4, which yields a quotient of 4 and a remainder of 1, these results can be checked by plugging into the relationship \(a = bq + r\), which simplifies back to 17 thus confirming the accuracy of the quotient and remainder.

Step 1: Identify the numbers in the problem

Considering for example the long division problem 17 divided by 4. Here \(a = 17\) (the number we are dividing), \(b = 4\) (the number we are dividing by), \(q\) is the quotient and \(r\) is the remainder.

Step 2: Carry out the long division

Divide 17 by 4, the quotient \(q = 4\) and the remainder \(r = 1\). We state the division as 17 divided by 4 equals 4 with a remainder 1.

Step 3: Use the Division Algorithm to verify

Substitute the identified numbers into the Division Algorithm, i.e. \(a = bq + r\). This forms the equation \(17 = 4 * 4 + 1\). Simplify the equation, if it is correct, then the quotient and remainder are accurate.

Step 4: Verification

When you simplify the equation, you find that it simplifies down to the original dividend (17 = 16 + 1). This confirms the quotient and the remainder are accurate.