Find all solution sets (x,y) for the equation:  x! = 20 y!
x and y are both positive integers.

There are two different answers for this equation (that I know of)

Let's explode 20! and look at it:

20! = 20x19x18x17x16x15x14x13x12x11x10x9x8x7x6x5x4x3x2x1

What do we get if we strip the 20 off of the beginning of 20!?

??! = 19x18x17x16x15x14x13x12x11x10x9x8x7x6x5x4x3x2x1

What did you end up with?

19! = 19x18x17x16x15x14x13x12x11x10x9x8x7x6x5x4x3x2x1

x! = 20 * 19!

What is x equal to?

There is a second answer.

Can we find a second factorial that has "20" first?

What is 20 equal to? It is equal to 2x10, 4x5, 5x4, 10x2

Do any of those look like the start of another factorial?

Yes, there is a second answer to your question.

Look at your first few factorials, explode them out like I did with 20! and see if "20" comes at the beginning of one of the first few. Hint, if you get past 6! and haven't found it, you've missed it.

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