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.

Explanatory Answer

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 p₁^a * p₂^b, 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.