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April 29th, 2014, 03:07 AM  #1 
Newbie Joined: Apr 2014 From: India Posts: 1 Thanks: 0  Euclid's division of algorithm
Hi, I am not able to understand how to apply the euclid's(a=b*q+r )division formula and solve the below problems. can you please help me solve the problem. 1. Prove that the product of 3 consecutive positive integers is divisible by 6? 2. For any positive integer N, prove that ( NÂ³  N ) is divisible by 6 ? Thanks!! 
April 29th, 2014, 03:42 AM  #2 
Senior Member Joined: Apr 2014 From: Greater London, England, UK Posts: 320 Thanks: 156 Math Focus: Abstract algebra 
Use the algorithm to show that any $b$ consecutive integers contains a multiple of $b$. Let the integers be $a+1,\,a+2,\,\ldots,\,a+b$. Write $a=bq+r$ where $0\leqslant r<b$. Now $0\leqslant r<b$ $\implies$ $0 < br\leqslant b$ $\implies$ $a<a+br\leqslant a+b$. Thus one of the numbers $a+1,\,a+2,\,\ldots,\,a+b1$ is equal to $a+br=(q+1)b$ i.e. it is divisible by $b$. Thus a sequence of $3$ consecutive integers contains a multiple of $3$. Also any sequence of $3$ consecutive integers must contain a sequence of $2$ consecutive integers and so must contain a multiple of $2$. Hence a product of $3$ consecutive integers is divisible by $3$ and also divisible by $2$; as $3$ and $2$ are relatively prime the product must be divisible by $3\times2=6$. Now to your questions: 1. The above result is true for any $3$ consecutive integers; in particular it is true for any $3$ consecutive positive integers. 2. $N^3N=(N1)\cdot N\cdot(N+1)$. Last edited by Olinguito; April 29th, 2014 at 04:10 AM. 

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algorithm, division, euclid 
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