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Factorial Product Poser (Posted on 2025-02-27)

Determine all possible pairs (x, y) of nonnegative integers that satisfy this equation:

x!=120*(y!)

Prove that there are no further pairs in consonance with the given conditions.

No Solution Yet Submitted by K Sengupta

Rating: 5.0000 (1 votes)

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Solution

| Comment 1 of 5

Rewrite the problem as 120=x!/y!

Thus, 120 must be the product of consecutive numbers.

Since 5!=210

There are 5 consecutive numbers using (5,0)

and 4 consecutive numbers using (5,1)

Since 4*5*6=129 we have (6,3)

120 cannot be written as the product of two consecutive numbers.

A single number 120=120!/119! so the final solution is (120,119).

Posted by Jer on 2025-02-27 08:06:38

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