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

(D) 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.
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The answer should be E. 4 , because your explanation has missed the possibility of 20² × 20² . It is not mentioned that the numbers should be distinct.

20² × 20² = 400 × 400 = 160,000 , NOT 400.