Let a be a positive integer. If n is divisible by 2ª and n is also divisible by 3²ª, then it is possible that n is NOT divisible by

A. 6
B. 3 × 2ª
C. 2 × 3²ª
D. 6ª
E. 6²ª

(E) If n is divisible by 2ᵃ and 3²ᵃ then it must be divisible by least common multiple of 2ᵃ and 3²ᵃ which equals 2ᵃ × 3²ᵃ.

Therefore the smallest possible value number n can take is 2ᵃ × 3²ᵃ which is less than answer choice (E), 6²ᵃ = 2²ᵃ × 3²ᵃ. A larger number can not be a divisor of a smaller one so 2ᵃ × 3²ᵃ is NOT divisible by 6²ᵃ. It means, that 6²ᵃ is not necessarily a divisor of n.

The right answer is choice (E).
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Why do we conclude from "that this number is smaller than choice (E)," that choice (E) is the right one?

We make a conclusion that choice (E) is the right one because we've shown that if n = 2ª × 3²ª (it fully satisfies the question statement) then it is not divisible by 6²ª simply because 6²ª is greater.

And furthermore I'd like to go over this question once again in details.
The question statement tells us that n is divisible by 2ª and 3²ª, where a is some positive integer. Therefore we can represent n as: n = 2ª × 3²ª × d, where d is some positive integer.

Note, that we consider n to be a positive integer for simplicity since divisibility of n or -n is the same.

We write the above mentioned formula for n because 2ª and 3²ª are relatively prime. If they where, for example, 10 and 15 (instead of 2ª and 3²ª) then n would equal 2 × 3 × 5 × d because 2 × 3 × 5 is the least common multiple of 10 and 15 while the least common multiple of 2ª and 3²ª is 2ª × 3²ª.

Let's return to our formula and consider all the answer choices:
n = 2ª × 3²ª × d

A. Is n divisible by 6?
Yes, because it is clearly divisible by 2 and 3.

B. Is n divisible by 3 × 2ª
Yes, because it is clearly divisible by 3 and 2ª is a multiplier in formula for n.

C. Is n divisible by 2 × 3²ª
Yes, because it is clearly divisible by 2 and 3²ª is a multiplier in formula for n.

D. Is n divisible by 6ª
Yes, because we can rewrite the formula for n:
n = 2ª × 3ª × 3ª × d and 6ª = (2 × 3)ª = 2ª × 3ª
So 2ª and 3ª are the multipliers in formula for n.

E. Is n divisible by 6²ª
It can be or it can be not. Let us divide n by 6²ª:
n / 6²ª = (2ª × 3ª × 3ª × d) / (2ª × 2ª × 3ª × 3ª) = d / 3ª
So n / 6²ª = d / 3ª

We clearly see that if d is divisible by 3ª then n is divisible by 6²ª. But if , let's say, d = 1 then n is not.

Therefore n MUST BE divisible by the choices (A) – (D) and NOT NECESSARILY by choice (E).

Choice (C), 2 × 3²ª, is much smaller than Choice (E), 6²ª = (2 × 3)²ª = 2²ª × 3²ª.

Note that notations (2 × 3)²ª and 2 × 3²ª mean two different numbers.

Choices (D) and (E) also differ.
6²ª = (6²)ª = 6ª × 6ª or 36ª. It is clearly greater than choice (D), 6ª.

Because, if n is divisible by 2ª and n is also divisible by 3²ª, then n MUST be divisible by 3 × 2ª (B). While the question asks us to choose a value that might NOT be a divisor of n.

In other words, we don't know what the real value of n is. It can be any value that satisfies "n is divisible by 2ª and n is also divisible by 3²ª". Choices A, B, C, D are divisors of any such value of n. While there is at least one possible value of n (e.g. 2ª × 3²ª) for which E, 6²ª, is NOT a divisor.

This is more a 600-700 question, though it might look harder. But the concept is very simple:

1. The smallest possible number, which is divisible by both 2ª and 3²ª is 2ª × 3²ª .
2. Choice E, 6²ª , is greater than 2ª × 3²ª , so it is definitely NOT a divisor of 2ª × 3²ª.

We have shown that at least one possible value of n (2ª × 3²ª) is NOT divisible by E. So E might NOT be a divisor of n.

The question tests the following abilities:
- dealing with powers
- knowing what factorization is and how it is connected to divisibility.


ANOTHER APPROACH.
Take a look at the answer choices. Exactly one of them must be the correct answer.

Choice E is divisible by any other answer choice. So if E is a divisor of n, then all the other answer choices are divisors as well.

Thus E cannot be the divisor of all possible values of n, because in this case all the answer choices would be the divisors and we would have NO correct answer.

A. 6
B. 3 × 2ª
C. 2 × 3²ª
D. 6ª
E. 6²ª