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Sum of factors of even positive and perfect square number N

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Sum of factors of even positive and perfect square number N

by goelmohit2002 » Wed Aug 05, 2009 10:12 am
Hi All,

Can some one please tell what is the "Sum of all the positive factors of a even positive and perfect square number N"...e.g. N = 36, 64, 100

so what is the sum of all the positive factors of N.

is it
a) even
b) odd...

If we see the same in the above examples, it comes to be odd....

But can someone please tell how to prove the same mathematically that it is always odd.

Thanks
Mohit
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Re: Sum of factors of even positive and perfect square numbe

by quant-master » Wed Aug 05, 2009 10:42 am
goelmohit2002 wrote:Hi All,

Can some one please tell what is the "Sum of all the positive factors of a even positive and perfect square number N"...e.g. N = 36, 64, 100

so what is the sum of all the positive factors of N.

is it
a) even
b) odd...

If we see the same in the above examples, it comes to be odd....

But can someone please tell how to prove the same mathematically that it is always odd.

Thanks
Mohit
If Perfect Square N is odd: An odd number will not have even factors and since square number will have odd number of factors. Sum of all the odd factors will be odd

If Perfect Square N is even: This is bit tricky! We have to find number of even and odd factors. Lets say 100. 100=22*52.

To find number of odd factors in 100 take 22 out of 100 (since odd factors will not have even multiple) we are left with 52 which will have 3 factors (2+1). Hence we have 3 odd factors in 100. Number of factors in 100 is (2+1)*(2+1) = 9 (I am adding one to the exponent of 2 and 5 and this is the standard method of finding number of factors)

We found that total number of factors is 9 and there are 3 odd factors hence we will have 6 even factors (9-3). Now 6 even factors added together will be an even number and 3 odd factors added together will be an odd number Hence Even+Odd = Odd

Hope this is clear

Thanks,
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https://gmat-quants.blocked - My Blog Updated almost daily with new quant fundas. Find collection of quants question in my blog
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by goelmohit2002 » Wed Aug 05, 2009 10:51 am
Awesome QM....this is real awesome !!!!!

Thanks a lot !!!

Really helpful.
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by quant-master » Wed Aug 05, 2009 10:54 am
goelmohit2002 wrote:Awesome QM....this is real awesome !!!!!

Thanks a lot !!!

Really helpful.
Thanks...added this concept to my blog now :D

Quant-Master
https://gmat-quants.blocked - My Blog Updated almost daily with new quant fundas. Find collection of quants question in my blog
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