2013 AMC 10B Problem 5

Factor the expression:
$$2a - ab = a(2 - b).$$
Here $a, b \in \{1, 2, 3, 4, 5\}$. The factor $a$ is positive, so the product is most negative when $2 - b$ is as negative as possible and $a$ is as large as possible.

Take $b = 5$, giving $2 - b = -3$, and $a = 5$:
$$5 \cdot (2 - 5) = -15.$$

Any other choice gives $a(2-b) \ge 5 \cdot (-3) = -15$, since $a \le 5$ and $2 - b \ge -3$.

The answer is $\boxed{\textbf{(B)}\ -15}$.

Factoring turns a two-variable expression into a product whose sign and size are easy to control: push each factor to its extreme in the direction that helps. With tiny ranges, checking the boundary values is enough, but the factored form explains why the boundary is optimal.

• Allowing a or b to equal 6, which gives 5(2 - 6) = -20, choice (A); the integers must be strictly less than 6.
• Taking $a$ small to "reduce" the expression, forgetting that a negative quantity gets smaller when multiplied by a larger positive number.