06/06/2020 (2011: National Team Round, Problem 7)

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

A: 45

 There are no restrictions given, so that means {x, y} is different from {y, x} (if x isn't equal to y). We can also have x = y, because of the fact that there are no restrictions. We can count them out as shown below **I am ordering them by the x-value in descending order (vertically, top to bottom), and every line has the same x-value. I am also ordering them by y-value in ascending order (horizontally, left to right).**

{400, 1}, {400, 2}, {400, 4}, {400, 5}, {400, 8}, {400, 10}, {400, 16}, {400, 20}, {400, 25}, {400, 40}, {400, 50}, {400, 80}, {400, 100}, {400, 200}, {400, 400}

{200, 16}, {200, 80}, {200, 400}

{100, 16}, {100, 80}, {100, 400}

{80, 25}, {80, 50}, {80, 100}, {80, 200}, {80, 400}

{50, 16}, {50, 80}, {50, 400}

{40, 400}

{25, 16} {25, 80}, {25, 400}

{20, 400}

{16, 25}, {16, 50}, {16, 100}, {16, 200}, {16, 400}

{10, 400}

{8, 400}

{5, 400}

{4, 400}

{2, 400}

{1, 400}

There are 45 pairs, so the answer is 45.