If N is a positive integer, such that the total number of factors . .

Question

If N is a positive integer, such that the total number of factors of N is q, and the sum of powers of different prime factors as expressed in the prime factorization form of N is p, is  p^q  even?
 (1) q is odd
(2) p is even

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ArunSharma12 · Accepted Answer

Given: N > 0
 N= x^a*y^b*z^c... 
(a+1)(b+1)(c+1)... = q
a + b + c + ... = p
 p^q = even ?

statement 1: q is odd
this implies the N is perfect square and all the coefficients of prime factors of N are even, thus P is even.
 p^q = even ; sufficient

statement 2: p is even
 q can be 0 or any other positive integer 
q can not be 0 as it is the total number of factors of N.
 p^q = even 
sufficient

Ans: A

Ans: D

sauravleo123 · Answer

N= x^a y^b z^c
(a+1)(b+1)(c+1)=q
(a+b+c)=p
Statement 1:
q=odd
a,b,c all have to be even
So p^q=even
Sufficient
Statement 2:
p=even
q cannot be 0 as q is the number of factors of a pos int N
So p^q will always be even if p is even
Sufficient
D

CrackverbalGMAT · Answer

N is a positive integer i.e. N = {1,2,3,4,…..}. The total number of factors of N is q and the sum of the powers of different prime factors is p.
By the language used in the question statement (prime factorization of N), we may conclude that N is a composite number. Therefore,  N cannot be 1 since 1 cannot be prime factorized .

The standard representation of a composite number is N =  a^x  *  b^y  *  c^z  *………, where,

a, b, c, etc represent the prime factors in the prime factorization of N AND,

x, y, z etc represent the positive integral powers of the prime factors of N.

For such a number, the total number of factors = (x+1) (y+1) (z+1)……

Therefore, q = (x+1)(y+1)(z+1)………
Also, p = x + y + z + ……

From statement I alone, q is odd. This means that the number of factors of N is odd.

Any number which has an odd number of factors is a Perfect Square .

Thus,  statement I alone says that the number N is a perfect square . Therefore,  the powers of the all the prime factors of N has to be even . Therefore, p HAS to be even since it is the sum of even numbers.  p^q  HAS to be even. 
Statement I alone is sufficient. Answer options B, C and E can be eliminated. Possible answer options are A or D.

From statement II alone, p is even. The sum of the powers of the distinct prime factors is even. 
Therefore,  p^q  will always be even since the base is even. Statement II alone is sufficient to answer the question with a definite YES. Answer option A can be eliminated.

The correct answer option is D .

Hope that helps!

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If N is a positive integer, such that the total number of factors . .

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