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^55n 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: 551 
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: 541 Thanks: 82 
$\displaystyle k=n^{55} n$ 
November 15th, 2018, 09:36 AM  #4 
Math Team Joined: May 2013 From: The Astral plane Posts: 2,205 Thanks: 901 Math Focus: Wibbly wobbly timeywimey stuff.  
November 15th, 2018, 09:50 AM  #5 
Senior Member Joined: May 2016 From: USA Posts: 1,310 Thanks: 551  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: 20,759 Thanks: 2138 
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  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
ABC is a three digit number. If ABC is divisible by 3, is CBA divisible by 3?  Tangeton  Number Theory  5  April 13th, 2016 12:01 PM 
How to find if a number is divisible by n  YuvalM  Elementary Math  2  October 27th, 2015 06:28 AM 
Divisible by 8  misosoup  Number Theory  7  June 2nd, 2015 11:34 AM 
HELP! probability that a 7 digit number is divisible by 7?  lincoln40113  Advanced Statistics  18  October 14th, 2013 11:52 PM 
Number system  proving 9 digit number not divisible by 5  sachinrajsharma  Number Theory  7  April 29th, 2013 05:49 AM 