PP problem

[Sephornia]

Which of the following cannot be a factor of 2^n x 3^k , where n and k are positive integers.

a) 6
b) 8
c) 27
d) 42
e) 54

Somehow I can't figure this one out. Correct me if I am doing something wrong.

First of all, I figured that as long as n and k are real integers, 2^n will produce an even number and 3^k will produce an odd number. From number theory, it is obvious that multiplying an even integer and an odd integer will produce an even number.

Given that, I thought that the answer would be C. But the answer is actually D. Anyone know why?

[FilleNouvelle]

Try writing all the answers in the form 2^n x 3^k

A) (2^1)(3^1), works.
B) (2^3)(3^0), works.
C) (2^0)(3^3), works.
D) 42= 6x7 = 2x3x7, doesn't work.
E) (2^1)(3^3), works.

If they're not readily apparent to you (like E wasn't to me), do out the prime factorizations. And don't forget any number raised to the 0 power is 1!

[Sephornia]

Oh right! I completely forgot about raising the numbers to the power of 0. Well regardless, your explanation makes a lot more sense. Thanks!

[vvv134]

i dont think you can raise to power 0 since it is mentioned n,k are positive integers only.

6 can be a factor of n and k being same number.
8 is a factor of n=3 and k=any number.
27 is a factor of k=3,n=any number.
54 is a factor of k=3,n=any even number.
42 cannot be factor since there is no power of 3 or 2 divisible by 7.