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April 29th, 2014, 03:07 AM   #1
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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 ( N³ - N ) is divisible by 6 ?

Thanks!!
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April 29th, 2014, 03:42 AM   #2
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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 $0\leqslant r<b$ $\implies$ $0 < b-r\leqslant b$ $\implies$ $a<a+b-r\leqslant a+b$. Thus one of the numbers $a+1,\,a+2,\,\ldots,\,a+b-1$ is equal to $a+b-r=(q+1)b$ 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. $N^3-N=(N-1)\cdot N\cdot(N+1)$.

Last edited by Olinguito; April 29th, 2014 at 04:10 AM.
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