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Thursday, January 17, 2008

Proof by contradiction, and a few problems

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

3 comments:

Efrique said...

5) Take n=3

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

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

Efrique said...

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

Filip Jekic said...

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