How many ordered pairs of positive integers (x,y) satisfy lcm(x,y) = 400?

Source: MathMovesU MATH CHALLENGE PROBLEM - 2011/12/21

Level: Senior

Since lcm(x, y) = 400 = (2^4)*(5^2), so x and y must be of the forms x = (2^a)*(5^b) and y = (2^c)*(5^d), where max(a, c) = 4, max(b, d) = 2, 0 <= a, c <= 4, and 0 <= b, d <= 2. If 0 <= a <= 3, then c must be 4, and this results in 4*1 = 4 possible values of a and c. If a = 4, then 0 <= c <= 4, and this results in 1*5 = 5 possible values for a and c. If 0 <= b <= 1, then d must be 2, and this results in 2*1 = 2 possible values for b and d. If b = 2, then 0 <= d <= 2, and this results in 1*3 = 3 possible values for b and d. In all, we have (4+5)*(2+3) = 45 possible values for a, b, c, and d. Hence 45 ordered pairs of positive integers (x,y) satisfy lcm(x,y) = 400.