Analyzing the Condition: a - b is Even

The condition that the difference between two integers, $a - b$, is even implies that $a$ and $b$ must have the same parity. This means either both $a$ and $b$ are even, or both $a$ and $b$ are odd.

Evaluating the Options

We need to find the expression that is always even, given that $a$ and $b$ have the same parity.

• Option 1: $ab$
  • If $a$ and $b$ are both odd: $a * b$ = odd * odd = odd.
  • Since this case results in an odd number, $ab$ is not always even.
• Option 2: $a² + b² + 1$
  • If $a$ and $b$ are both even: $a²$ is even, $b²$ is even. So, $even + even + 1$ = odd.
  • If $a$ and $b$ are both odd: $a²$ is odd, $b²$ is odd. So, $odd + odd + 1$ = $even + 1$ = odd.
  • This expression is always odd.
• Option 3: $a² + b + 1$
  • If $a$ and $b$ are both even: $a²$ is even. So, $even + even + 1$ = odd.
  • If $a$ and $b$ are both odd: $a²$ is odd. So, $odd + odd + 1$ = $even + 1$ = odd.
  • This expression is always odd.
• Option 4: $ab - b$
  • We can factor this expression as $b(a - 1)$.
  • Case 1: Both $a$ and $b$ are even. Let $a = 2k$ and $b = 2m$ for some integers $k, m$. $ab - b$ = $(2k)(2m) - 2m$ = $4km - 2m$ = $2(2km - m)$. This is always even.
  • Case 2: Both $a$ and $b$ are odd. Let $a = 2k + 1$ and $b = 2m + 1$ for some integers $k, m$. $a - 1$ = $(2k + 1) - 1$ = $2k$ (which is even). $ab - b$ = $b(a - 1)$ = $odd * even$ = even.
  • Since the expression is even in both possible cases (both even or both odd), $ab - b$ must always be even.

Conclusion

Based on the analysis of parity, the expression $ab - b$ is the only one guaranteed to be even when $a - b$ is even.