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

= 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
 * $$n(n+33)(n+46)(n+92)(n+74) \equiv n(n+3)(n+1)(n+2)(n+4) \mod 5$$,
 * 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)