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Hard to understand this one

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by yogami » Sat Jul 18, 2009 4:11 pm
Guys I am having trouble getting a better explanation with this one. I wasn't satisfied with the one given by manhattan guys. Can someone break it down to me in simpler steps pls?

Is the positive integer N a perfect square?

(1) The number of distinct factors of N is even.
(2) The sum of all distinct factors of N is even.

OA l8r
200 or 800. It don't matter no more.
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Source: — Data Sufficiency |
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Re: Hard to understand this one

by El Cucu » Sat Jul 18, 2009 4:50 pm
yogami wrote:Guys I am having trouble getting a better explanation with this one. I wasn't satisfied with the one given by manhattan guys. Can someone break it down to me in simpler steps pls?

Is the positive integer N a perfect square?

(1) The number of distinct factors of N is even.
(2) The sum of all distinct factors of N is even.

1) Sufficient as if the number of factors is even N is not a perfect square. Ej 4 is 2^2 but 4 has 1,2,and 4 as factors (3factors).
2) Sufficient. 49 has as factors 49,1,7 the sum is odd, 16 has as factors 1,2,4,8,16 and the sum is also odd, so N is not a perfect square.
Answ should be D.
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Re: Hard to understand this one

by PussInBoots » Sat Jul 18, 2009 5:19 pm
yogami wrote:(1) The number of distinct factors of N is even.
SUFFICIENT. N is not perfect square. Any perfect square has odd number of factors (1 .... sqrt ... itself, ... and ... have equal number of factors)
8: 1, 2, 4, 8 not perfect square
yogami wrote:(2) The sum of all distinct factors of N is even.
N is perfect square and odd, all of its factors are odd and there are odd numbers of them. The sum has to be odd.
N is perfect square and even. number + sqrt + 1 = odd number. Idk the rest but it has to be D, lol
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Re: Hard to understand this one

by goelmohit2002 » Wed Aug 05, 2009 7:29 am
PussInBoots wrote: N is perfect square and odd, all of its factors are odd and there are odd numbers of them. The sum has to be odd.
N is perfect square and even. number + sqrt + 1 = odd number. Idk the rest but it has to be D, lol
why are we ignoring the numbers like 36 = square of 36

it's factors sum is not same as 36 + 1 + 6

but in fact 1 + 2+ 3+ 4+ 6+ 9+ 12+18+36

Basically how to prove the "N is perfect square and even." case ?

Can some one please tell ?
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by goelmohit2002 » Wed Aug 05, 2009 7:31 am
Can someone please tell why are we ignoring the negative factors....why the answer should not be E ?

IMO all the numbers have both positive and negative factors...

e.g. 4 has following factors: -4, -2, -1, 1, 2, 4

So all the numbers will have even number of factors(except zero) and sum of factors is always zero.....i.e. even...

Please tell what I am missing here ?
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by tohellandback » Wed Aug 05, 2009 10:52 pm
IMO D

the underlying concept:
any number N is represented as

N=a^m*b^n*c^p....
where a,b,c are prime.
total number of distinct factors: (m+1)(n+1)(p+1)..
also, if the N is the square of a number, then m,n,p..must be even

(1) The number of distinct factors of N is even.
means.. (m+1)(n+1)(p+1)..is even
so at least one of the terms (m+1),(n+1)...must be even
so at least one of m,n,o..must be odd. so number N is not a square.
sufficient

2)plugin numbers
for 5, 1+5=6-even
6, 1+2+3+6-even
4, 1+2+4-odd
25-1+5+25-odd
sufficient
The powers of two are bloody impolite!!
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