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Factorial and a Square (Posted on 2008-02-23) Difficulty: 2 of 5
If a and p are positive integers, then solve for (a,p):

See The Solution Submitted by Praneeth    
Rating: 2.0000 (4 votes)

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Solution easy, when you know how. | Comment 3 of 9 |

Let P be the largest prime less than p. Then P divides p!, hence P also divides a^2. But for prime P to divide a^2, it must also divide a, that is, P^2 divides p!, so that p must be >= 2P. However, we know that for any number x >= 1, there is a prime between x and 2x.

It follows that the sole solution is a=p=1.

Edited on February 24, 2008, 10:24 am
  Posted by FrankM on 2008-02-24 10:21:29

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