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Powerful Divisor (Posted on 2014-05-01) Difficulty: 3 of 5
Let us consider the expression MM+1, where M is a positive integer.

It can be verified that M=3 is the least value for which 22 divides MM+1.

Given that n is a positive integer, find the least value of M (in terms of n) for which MM+1 is divisible by 2n.

No Solution Yet Submitted by K Sengupta    
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Solution solution | Comment 4 of 8 |
For a given value of n the least value of M is 2n-1.

(2n-1)^(2n-1) + 1
Expand the binomial.  (Since the exponent 2n-1 is odd the last term will be -1 and the + 1 will cancel this)
(2n)^(2n-1) - (2n-1)(2n)^(2n-2) + ... + (2n-1)(2n) - 1 + 1
so every term left is divisible by 2n.

In general the largest power of 2 that divides M+1 is the largest power of 2 that divides MM+1.

  Posted by Jer on 2014-05-01 16:38:40
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