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?

Solution Steps

• $a$ 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.

• 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.

• 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.

Final Answer