2013 AMC 10B Problem 5

1. Both $a$ and $b$ can only be $1, 2, 3, 4$ or $5$.
2. Both parts of $2a - ab$ contain $a$, so pull it out:
$$2a - ab = a(2 - b).$$
A product of two numbers is easier to control than a difference.
3. The factor $a$ is always positive. To make the product as small as possible (as negative as possible), the other factor $2 - b$ should be negative, and as negative as it can get. Here are its values:

The most negative is $-3$, when $b = 5$.
4. Now multiply $-3$ by the largest $a$ allowed, which is $5$: $5 \times (-3) = -15$.
5. Nothing can do better. The factor $a$ is at most $5$, and $2 - b$ is never below $-3$, so the product can never go below $5 \times (-3) = -15$.

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

Factoring turned an expression with two letters into a product. For a product, you can reason about each factor separately: choose the sign you need, then push the size as far as it goes. Always read the limits carefully; "less than $6$" means $6$ itself is not allowed.

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