# DS Number Properties - Divisibility - Prime Divisors

This is an interesting Data Sufficiency question that tests your understanding of divisibility, indices and prime factors.

Question
If n is an integer, is n3 divisible by 54?
1. n2 is divisible by 6.
2. n3 is divisible by 36.

Correct Answer : Choice D. Each statement is independently sufficient to answer the question.

For "is" questions in DS, we need to answer with a clear Yes or a clear No. If the data in the statements does not lead to arriving at a definite Yes or No, the data is insufficient.

We know from the question stem that n is an integer.

I. Let us look at statement 1 alone : n2 is divisible by 6.
If n2 is divisible by 6, then n2 is divisible by both 2 and 3 - the prime factors of 6.
But, we know that n is an integer.
Therefore, n2 will be of the form p1a * p2b, where p1 and p2 are prime factors of n and a and b are even.
Hence, we can deduce that when  n2 is expressed in terms of its prime factors, the power of 2 and 3 in it will be even.
So,  n2 will be divisible by both 22 and 32. Hence, n is divisible by 2 and 3.

If n is divisible by 2 and 3, then n3 will be divisible by 23 and 33 or by 216.
If n3 is divisible by 216, it will certainly be divisible by any factor of 216 - and therefore by 54.

Statement 1 alone is sufficient.
Answer is either choice A or choice D.

II. Let us look at statement 2 alone :  n3 is divisible by 36.
For integer n, when n3 is expressed in terms of its prime factors, the powers of the prime factors will be multiples of 3.
So, if n3 is divisible by 36 or 22 * 32, we can deduce that n3 is actually divisible by 23 and 33 as the power of 2 and 3 should be a multiple of 3.
If n3 is divisible by 23 and 33, it is divisible by 23 and 33 or by 216.

Statement 2 alone is sufficient.

Because each of these statements is independently sufficient, Choice D is the answer.