Sample question from ETS website

[scheng75]

This is a sample Miscellaneous Problems that Require Developing Problem-Solving Strategies of difficulty level 5 from the ETS website when I checked my results:

Which of the following CANNOT be a factor of 2n . 3k, where n and k are positive integers?

(A)6(B)8(C)27(D)42(E)54
Please help me with a quick way of solving it!

[danielman]

This is a sample Miscellaneous Problems that Require Developing Problem-Solving Strategies of difficulty level 5 from the ETS website when I checked my results:

Which of the following CANNOT be a factor of 2n . 3k, where n and k are positive integers?

(A)6(B)8(C)27(D)42(E)54
Please help me with a quick way of solving it!

I guess the question is about the number (2^n)(3^k)

Otherwise it makes no sense

If this is the question, the answer is D because 42 has a factor 7 and the number (2^n)(3^k) doesn't.

[scheng75]

Yes, it is 2^n and 3^k. Answer is 42. So, you look for the prime factor that the number has but multiples of 2 and 3 do not?

[danielman]

Yes, it is 2^n and 3^k. Answer is 42. So, you look for the prime factor that the number has but multiples of 2 and 3 do not?

the reasoning is as follows: when an integer A is a factor of another integer B? when there is another integer N such that: B = N*A

(2^n)(3^k) = N*A

every option but 42 satisfy this condition, for instance, for A = 6

(2^n)(3^k) = N*6 -> N = (2^(n-1)) * (3^(k-1))

but for A = 42 such N doesn't exist as A = 2*3*7:

N = [2^(n-1) * 3^(k-1)] / 7 -> N is not an integer

[777]

k (x) = 2^n(3^k)

So we can infer that x is either a multiple of 2 or 3 or both. This leaves out 42 because of it has a factor of 7.