Bunuel wrote:

The product of the squares of two positive integers is 400. How many pairs of positive integers satisfy this condition?

A. 0

B. 1

C. 2

D. 3

E. 4

Kudos for a correct solution.

800score Official Solution:400 = 2 × 2 × 2 × 2 × 5 × 5

Combine the prime factors in pairs.

400 = (2 × 2) × (2 × 2) × (5 × 5)

Now brake the factorization into two parts, each one will be a square.

The possible combinations are:

400 = (2 × 2) × [(2 × 2) × (5 × 5)]

400 = [(2 × 2) × (2 × 2)] × (5 × 5)

But don't forget that 400 = 1 × 400, where 1 = 1². So we also have:

400 = (1 × 1) × [(2 × 2) × (2 × 2) × (5 × 5)]

Thus all the possible combinations of the factors that make the product of two squares are the following:

1² × 20² = 400

2² × 10² = 400

4² × 5² = 400

There are three possible pairs that fit the criterion.

The correct answer is D.

And then approach it like this? The number of factors in 20 are 6, since we are looking for pair of factors, we divide de total number of factor by 2 and get 3 as a result.