2015 AMC 12B Problem 6

Question

Back in

1930,

Tillie had to memorize her multiplication facts from

0 \times 0

through

12 \times 12.

The multiplication table she was given had rows and columns labeled with the factors, and the products formed the body of the table. To the nearest hundredth, what fraction of the numbers in the body of the table are odd?

0.21

0.25

0.46

0.50

0.75

Answer: A

Concepts:parity basic counting

Difficulty rating: 1290

Small hint:

A product is odd only when both factors are odd

Big hint:

Count the odd numbers from

0

to

12,

then compare with the

13 \times 13

entries

Solution:

The body has

13 \times 13 = 169

entries. A product is odd exactly when both factors are odd.

There are

6

odd numbers among

0, 1, \ldots, 12,

giving

6 \times 6 = 36

odd entries. The fraction is

\dfrac{36}{169} = 0.213\ldots \approx 0.21.

Thus, the correct answer is A.