Problem 58 - College Algebra

Question

In your own words, state the Division Algorithm.

Step-by-Step Solution

Answer

The Division Algorithm in mathematics asserts that for any two integers \(a\) and \(b\) (with \(b\) not zero), there exist unique integers \(q\) and \(r\) such that \(a = bq + r\), where \(0 \leq r < |b|\). Here \(a\) is the dividend, \(b\) is the divisor, \(q\) is the quotient, and \(r\) is the remainder.

Step 1: Understand the Division Algorithm

The Division Algorithm is not an algorithm in the traditional sense but rather a theorem in number theory. It states that given any two integers \(a\) and \(b\) (where \(b\) is not zero), 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.

Step 2: Breakdown of the formula

In the equation \(a = bq + r\), we are saying that our number \(a\) can be expressed as the divisor (\(b\)) multiplied by the quotient (\(q\)), plus the remainder (\(r\)). This expression covers all possible scenarios of integer division. The condition \(0 \leq r < |b|\) ensures that the remainder is always a non-negative integer and smaller than the absolute divisor.

Step 3: Give Examples

Let's illustrate this using some random numbers. Say we have \(a = 42\) and \(b = 5\), carrying out a division we have the quotient \(q = 8\) and remainder \(r = 2\). This satisfies our division algorithm since \(42 = 5 * 8 + 2\). And indeed, our remainder is greater than or equal to 0 and less than 5.