Testwiki:Reference desk/Archives/Mathematics/2013 February 3

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February 3

divisibility

how do i show that those number devide by 10: /10 ?= \mathbb {N}. $\frac{[n(n+33)(n+46)(n+92)(n+74)]}{10}=\mathbb{N}$

thanks --84.110.35.108 (talk) 20:42, 3 February 2013 (UTC)— Preceding unsigned comment added by 84.110.35.108 (talk) 20:41, 3 February 2013 (UTC)

You need to check that the numerator is divisible by both 2 and 5 no matter what n is. To do this you only need to check all the numbers modulo 2 and 5. Dmcq (talk) 23:29, 3 February 2013 (UTC)

By the way it is always divisible by 3 as well so you could have 30 on the bottom. However it is not always divisible by 4. Dmcq (talk) 11:50, 4 February 2013 (UTC)

In more detail, we have that

$n(n+33)(n+46)(n+92)(n+74)\equiv n(n+3)(n+1)(n+2)(n+4)mod5$,

and one of n, n+1, n+2, n+3, and n+4 must be divisible by 5. A similar calculation works (mod 2). If x is 0 (mod 2) and (mod 5), then it must be 0 (mod 10). See modular arithmetic for more information. —Anonymous Dissident^Talk 21:35, 4 February 2013 (UTC)

I think this is rather obviously homework, is it not? Surely we should have been more careful. IBE (talk) 10:20, 5 February 2013 (UTC)