It is known that m^2 + n^2 + m is divisible by mn for some positive integers m and n.Prove that m is a perfect square.

Proof to this problem is proof by contradiction.If you can`t solve this one, try solving one of these:

1.Prove that there is no largest real number.

2.Prove that there is no largest prime number.

3.5 children have picked up 9 flowers.Prove that at least 2 of them picked up the same number of flowers.

4.Prove that (n-1)(n)(n+1) is divisible by 6.

5.Prove that (n-2)(n-1)(n)(n+1)(n+2) is divisible by 2*3*4*5.

6.Prove that if n divides (n-1)! + 1 then n is a prime.

3! = 1*2*3

## Thursday, January 17, 2008

### Proof by contradiction, and a few problems

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## 3 comments:

5) Take n=3

(n-2)(n-1)(n)(n+1)(n+2) = 5! = 120

is not divisible by 4*9*4*5.

I expect you wanted to say it's divisible by 2*3*4*5

You`re right.I wanted to post (n-3)(n-2)...(n+3) problem.Ty for noticing it.

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