2013 AMC 10B Problem 5

P1-10~2 minPrint

Algebraic Manipulation Absolute Value & Inequalities Consider the largest/smallest element or boundary case Test small/specific values or special cases to find or verify the answer

Positive integers $a$ and $b$ are each less than $6$ . What is the smallest possible value for $2 \cdot a - a \cdot b$ ?

Show solution without answering

Key insight

Write 2a - ab as a(2 - b). Use b = 5 to make (2 - b) as negative as possible, then the largest a, which is 5.

Solution

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:

$b$	$1$	$2$	$3$	$4$	$5$
$2 - b$	$1$	$0$	$-1$	$-2$	$-3$

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

Why this works

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.

Concise solution

A shorter write-up, the way contest solutions are usually written. It may take a different route from the solution above.

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

Review in the AoPS books

Introduction to Algebra, Chapter 11: Special Factorizations (§11.2, §11.3, §11.5)
Introduction to Algebra, Chapter 20: Special Functions (§20.2, §20.5)
Introduction to Algebra, Chapter 22: Special Manipulations
Introduction to Algebra, Chapter 15: More Inequalities (§15.1, §15.3, §15.4)