Problem 48 - College Algebra

Question

In your own words, state the Division Algorithm.

Step-by-Step Solution

Answer

The Division Algorithm states that for given two integers 'a' and 'b' with 'b' not equal to zero, there exist unique integers 'q' and 'r' such that 'a' equals to 'b' times 'q' plus 'r' where 'r' is greater or equal to 0 and less than 'b'.

Step 1: Understanding the Division Algorithm

The Division Algorithm states: For any given integers a and b, with b > 0, there exist unique integers q and r satisfying the equation \(a = bq + r\) where \(0 \leq r < b\). This theorem basically is describing the process of long division.

Step 2: Describing the Division

In the equation \(a = bq + r\), 'a' is the dividend, 'b' is the divisor, 'q' is the quotient and 'r' is the remainder. When we divide 'a' by 'b', we get a quotient 'q' and a remainder 'r'.

Step 3: Discussing Uniqueness and Range of Remainder

The Division Algorithm also tells us two things about the quotient and the remainder: Firstly, the quotient 'q' and the remainder 'r' are unique for given 'a' and 'b'. Secondly, the remainder 'r' always satisfy the property that \(0 \leq r < b\), meaning the remainder yielded from division will always be a non-negative integer and less than the divisor 'b'.