If n is an integer, is n odd? (1) n/3 is divisible by 3 (2)

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.
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Since we have 1 variable (\(n\)) and 0 equations, D is most likely to be the answer. So, we should consider each condition on its own first.

Condition 1)

We have an expression \(\frac{n}{3} = 3k\) for some integer \(k\).
Then we have \(n = 3^2k\).
If \(k = 1\), then \(n = 3^2 = 9\), \(n\) is an odd number and the answer is 'yes'.
If \(k = 2\), then \(n = 3^2 \cdot 2 = 18\), \(n\) is an even number and the answer is 'no'.

Since condition 1) does not yield a unique solution, it is not sufficient.

Condition 2)

Assume \(n = p^a \cdot q^b \cdot r^c\) where \(p, q\) and \(r\) are different prime numbers.
Then the number of factors of \(n\) is \((a+1)(b+1)(c+1)\).

If \(p, q\) and \(r\) are odd prime numbers, then \(2n = 2^1 \cdot p^a \cdot q^b \cdot r^c\) and the number of factors of \(2n\) is \((1+1)(a+1)(b+1)(c+1) = 2(a+1)(b+1)(c+1)\) which is twice as many factors as \(n\).

Assume \(p = 2\) which means one of prime factors of \(n\) is \(2\).
Then \(2n = 2p^a \cdot q^b \cdot r^c = 2^{a+1} \cdot q^b \cdot r^c\) and the number of factors of \(2n\) is \((a+2)(b+1)(c+1)\).
We have \((a+2)(b+1)(c+1) = 2(a+1)(b+1)(c+1)\) or \(a+2 = 2a + 2\). Then we have \(a = 0\) and n is an odd number.

Since condition 2) yields a unique solution, it is sufficient.


Therefore, D is the answer.

If the original condition includes “1 variable”, or “2 variables and 1 equation”, or “3 variables and 2 equations” etc., one more equation is required to answer the question. If each of conditions 1) and 2) provide an additional equation, there is a 59% chance that D is the answer, a 38% chance that A or B is the answer, and a 3% chance that the answer is C or E. Thus, answer D (conditions 1) and 2), when applied separately, are sufficient to answer the question) is most likely, but there may be cases where the answer is A,B,C or E.