2013 AMC 10B Problem 5

Question

Positive integers

a

and

b

are each less than

6.

What is the smallest possible value of the following expression?

2 \cdot a - a \cdot b

-20

-15

-10

0

2

Factor the expression as

a(2-b)

Make

2-b

as negative as possible while making

a

large

The expression is

2a-ab=a(2-b)

.

To make it as small as possible, choose

b

as large as possible so that

2-b

is most negative, and choose

a

as large as possible. Since

a,b<6

, take

a=b=5

.

The minimum value is

5(2-5)=-15

.

Thus, the correct answer is B .

Problem 5 in other years