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# If N is a positive integer, not including N, how many factors does N

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Math Expert
Joined: 02 Sep 2009
Posts: 49320
If N is a positive integer, not including N, how many factors does N  [#permalink]

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03 Jul 2018, 10:58
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Difficulty:

95% (hard)

Question Stats:

32% (01:58) correct 68% (01:46) wrong based on 44 sessions

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If N is a positive integer, not including N, how many factors does N have?

(1) Not including N^2, N^2 has 4 factors.

(2) Not including 2N, 2N has 3 factors.

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Math Expert
Joined: 02 Aug 2009
Posts: 6799
If N is a positive integer, not including N, how many factors does N  [#permalink]

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03 Jul 2018, 11:15
1
1
If N is a positive integer, not including N, how many factors does N have?

(1) Not including N^2, N^2 has 4 factors.
so $$N^2$$ has 5 factors including itself.. ODD factors means square number
what is 5 = 1*5..
as the number of factors of a number in simplest form $$x^ay^b$$ is (a+1)(b+1)
so 1*5 means (0+1)(4+1).....
so $$N^2= (x^2)^2$$, where x is prime
so x is a prime number and $$x^2$$ has (2+1) or 3 factors
suff

(2) Not including 2N, 2N has 3 factors.
2N has total 4 factors...
2*2 means (1+1)(1+1)
N can be any other prime such that 2N = 2*3 or 2*5
then factors of 2*3 = 1,2,3,6 ....4
so N will be a prime number and there will be TWO factors
4 is also 1*4=(3+1)..
so N can be 2^2 such that 2*2^2=2^3..
then N = 2^2 means (2+1) = 3 factors
insuff

A
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1) Absolute modulus : http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
2)Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html
3) effects of arithmetic operations : https://gmatclub.com/forum/effects-of-arithmetic-operations-on-fractions-269413.html

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Re: If N is a positive integer, not including N, how many factors does N  [#permalink]

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05 Jul 2018, 13:55
Bunuel wrote:
If N is a positive integer, not including N, how many factors does N have?

(1) Not including N^2, N^2 has 4 factors.

(2) Not including 2N, 2N has 3 factors.

St1:-
Re-phrasing:-$$N^2$$ has 5 factors. Or, N is the square of a prime number.
No of factors of N excluding N is 2 . (1,$$\sqrt{N}$$)
Sufficient.
St2:-
Re-phrasing:-2N has 4 factors.
This hold true when 2N=$$2^1*3^1$$or $$2^1*5^1$$ or $$2^2$$ or $$2*(2^2)$$ etc.
So, we have N=$$\frac{2N}{2}$$=3 or 5 or 2 or $$2^2$$
So, no of factors of N :- 1 or 1 or 1 or 2 (excluding N)
Since no of factors have more than one value.
hence insufficient.
Ans. (A)
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Re: If N is a positive integer, not including N, how many factors does N &nbs [#permalink] 05 Jul 2018, 13:55
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