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.  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.
