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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 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 by step solution

01

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.
02

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.
03

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.
04

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.

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