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 November 15th, 2018, 06:06 AM #1 Member   Joined: Oct 2012 Posts: 78 Thanks: 0 Large number divisible Find the largest integer k such that k divides n^55-n for all integer n? Last edited by skipjack; November 16th, 2018 at 04:07 AM. November 15th, 2018, 07:32 AM #2 Senior Member   Joined: May 2016 From: USA Posts: 1,310 Thanks: 552 Are you sure that this is the entire question? $\text {Find the largest integer k that divides evenly into } x \text { given that } x = n^{55} - n \text { and } n \in \mathbb Z.$ If $|n| \le 1$, there is no largest integer that divides into x because x will equal 0. Otherwise, the largest integer that can divide evenly into x is |x|. Last edited by JeffM1; November 15th, 2018 at 07:40 AM. November 15th, 2018, 09:35 AM #3 Senior Member   Joined: Dec 2015 From: somewhere Posts: 721 Thanks: 97 $\displaystyle k=n^{55} -n$ Thanks from topsquark November 15th, 2018, 09:36 AM   #4
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Quote:
 Originally Posted by fahad nasir Find the largest integer k such that k divides n^55-n for all integer n?
Are you, perhaps, trying to find a function k(n)?

-Dan November 15th, 2018, 09:50 AM   #5
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Quote:
 Originally Posted by idontknow $\displaystyle k=n^{55} -n$ Actually, that does not work. Consider n = 1. Then

$1^{55} - 1 = 1 - 1 = 0.$

And 0 does not divide evenly into 0. And $n = -\ 2 \implies 2 > (-\ 2)^{55} - (-\ 2) \text { and } 2 \ | \ (-\ 2)^{55} - (-\ 2).$ Really stupid problem as posted.

Last edited by JeffM1; November 15th, 2018 at 09:57 AM. November 16th, 2018, 04:11 AM #6 Global Moderator   Joined: Dec 2006 Posts: 21,026 Thanks: 2257 n^55 - n is divisible by n - 1, n and n + 1, so it's divisble by 2 × 3 = 6. Hence the largest k is at least 6. Tags divisible, larg, large, number Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post Tangeton Number Theory 5 April 13th, 2016 12:01 PM YuvalM Elementary Math 2 October 27th, 2015 06:28 AM misosoup Number Theory 7 June 2nd, 2015 11:34 AM lincoln40113 Advanced Statistics 18 October 14th, 2013 11:52 PM sachinrajsharma Number Theory 7 April 29th, 2013 05:49 AM

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