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        riddles >> medium >> Prove 30 divide (ba^5 - ab^5)   
        
(Message started by: wonderful on Jun 11^th, 2008, 5:16pm)

Title: Prove 30 divide (ba^5 - ab^5)
Post by wonderful on Jun 11^th, 2008, 5:16pm

Given a, b are integers, prove that:
30 | (ba^5 - ab^5)

Have A Great Day!

Title: Re: Prove 30 divide a^(5b) - b^ (5a)
Post by Aryabhatta on Jun 11^th, 2008, 8:18pm

Something is not right...

a = 2, b =1, we get 30 divides 31...

Title: Re: Prove 30 divide a^(5b) - b^ (5a)
Post by wonderful on Jun 11^th, 2008, 8:23pm

Thanks Aryabhatta for pointing that out. It should be: 30 | (ba^5 - ab^5). I have revised the original post accordingly.

Have A Great Day!

Title: Re: Prove 30 divide (ba^5 - ab^5)
Post by Aryabhatta on Jun 11^th, 2008, 8:54pm

Ok.

The following seems to work:
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Use the fact that

n⁵ = n mod 2
n⁵ = n mod 3
n⁵ = n mod 5

We see that the given expression is ba - ab mod 2,3 and 5 and hence is divisible by 30.

[/hide]

Title: Re: Prove 30 divide (ba^5 - ab^5)
Post by wonderful on Jun 11^th, 2008, 8:59pm

Excellent!

Have A Great Day!