Questions: If a is an even positive integer and b is an odd positive integer, then which of the following expressions will NEVER be an integer?

Transcript text: If $a$ is an even positive integer and $b$ is an odd positive integer, then which of the following expressions will NEVER be an integer?

Solution

Solution Steps

Step 1: Understand the given conditions

$ a $ is an even positive integer, so $ a = 2k $ where $ k $ is a positive integer.
$ b $ is an odd positive integer, so $ b = 2m + 1 $ where $ m $ is a non-negative integer.

Step 2: Analyze the properties of even and odd numbers

The sum of an even and an odd number is odd: $ a + b = 2k + (2m + 1) = 2(k + m) + 1 $, which is odd.
The product of an even and an odd number is even: $ a \cdot b = 2k \cdot (2m + 1) = 2k(2m + 1) $, which is even.

Step 3: Identify expressions that will never be an integer

Since $ a $ is even and $ b $ is odd, expressions like $ \frac{a}{b} $ or $ \frac{b}{a} $ may not always be integers. Specifically:
- $ \frac{a}{b} = \frac{2k}{2m + 1} $ will not be an integer unless $ 2m + 1 $ divides $ 2k $, which is unlikely for arbitrary $ k $ and $ m $.
- $ \frac{b}{a} = \frac{2m + 1}{2k} $ will not be an integer unless $ 2k $ divides $ 2m + 1 $, which is impossible since $ 2k $ is even and $ 2m + 1 $ is odd.

Thus, expressions like $ \frac{a}{b} $ or $ \frac{b}{a} $ will never be integers under the given conditions.