1997 AMC 8 Problem 11

Question

Let

\boxed{N}

mean the number of whole number divisors of

N.

For example,

\boxed{3}=2

because

3

has two divisors,

1

and

3.

Find the value of

\boxed{\boxed{11}\times\boxed{20}}.

6

8

12

16

24

Use prime factorization to count divisors

Evaluate the inside boxes before the outside box

We know that

11

is prime, which means that it only has

2

divisors.

The prime factorization of

20

is

2^2 \cdot 5.

Recall that the number of divisors a number has is the product of all the exponents plus one in the prime factorization.

Here, that product would be

(2 + 1) (1 + 1) = 3 \cdot 2 = 6.

Then

2 \cdot 6 = 12.

We have the prime factorization of

12

is

2^2 \cdot 3.

This also has

(2 + 1) (1 + 1) = 3 \cdot 2 = 6

divisors.

Thus, A is the correct answer.

Problem 11 in other years