Question

Let $b$ be a positive integer and $a = b^2 - b$. If $b \ge 4$, then $a^2 - 2a$ is divisible by:

• A. 15
• B. 20
• C. 24
• D. All of these

Hint:

Factorize expressions to reveal consecutive integer patterns for divisibility analysis.

Answer 1

The Correct Option is D

$a = b(b-1)$, so $a^2 - 2a = a(a-2) = b(b-1)(b(b-1) - 2)$. For $b \ge 4$, factors include three consecutive integers (ensuring divisibility by 3), two even numbers (ensuring divisibility by 4), and one multiple of 5 over a range of $b$, hence divisible by LCM of 15, 20, and 24.