If a and b are integers and a – b is even, which of the following must always be even?
Options:
(A) ab
(B) a² + b² + 1
(C) a² + b + 1
(D) ab – b
Official Answer Key:
D
Step-by-Step Solution & Explanation
• a – b even means a and b have the same parity (both even or both odd).
• Case both even: ab – b = b(a–1); b even, so the product is even regardless of (a–1).
• Case both odd: a–1 is even, so b(a–1) is even regardless of b being odd.
• In both cases ab – b is even; the other options fail in at least one parity case (e.g. for a=b=1 (both odd): ab=1 odd, a²+b²+1=3 odd, a²+b+1=3 odd).
• Final Answer: (D) ab – b
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