Question

In the given question, there are two statements marked I and II. Decide which of the statements are sufficient to answer the question. Choose your answer from the given alternative.

Is p divisible by $168$ ?
(I) P is divisible by $14$.
(II) P is divisible by $12$.

• A. Statement I alone is sufficient to answer the problem.
• B. Statement II alone is sufficient to answer the problem.
• C. Statement I and II both are needed.
• D. Statement I and II both are not sufficient.

Answer 1

The correct option is D Statement I and II both are not sufficient.

The first step in this kind of problem is to determine what prime factors p needs in order to be divisible by 168. The prime factorization of 168 is $2\times 2\times 2\times 3\times 7$, so the question can be restated as follows

Are there at least three 2's, one 3, and one 7 in the prime box of p?

INSUFFICIENT:

Statement (1) tells you that p is divisible by 14, which is 2 $\times$ 7. Therefore, you know that p has at least a 2 and a 7 in its prime box. However, you do not know anything else about the possible prime factors in p, so you cannot determine whether p is divisible by 168. For example, p could equal $2\times 2\times 2\times 3\times 7=168$, in which case the answer to the question would be, "Yes,pis divisible by 168." Alternatively, p could equal $2\times 7=14$, in which case the answer to the question would be, "No, p is not divisible by 168."

(2) INSUFFICIENT: Statement (2) tells you that p is divisible by 12, which is $2\times 2\times 3$. Therefore, you know that p has at least two 2's and one 3 in its prime box. However, you do not know anything else about p, so you cannot determine whether p is divisible by 168. For example, p could equal 168, in which case the answer to the question would be, "Yes,p is divisible by 168." Alternatively, p could equal 12, in which case the answer to the question would be "No, p is not divisible by 168.

"(1) AND (2) INSUFFICIENT: Combining the primes from statements (1) and (2), you seem to have three 2's, one 3, and one 7. That should be sufficient to prove that p is divisible by 168.

However, you cannot do this. Consider the number 84 : 84 is divisible by 14 and it is also divisible by 12. Therefore, following from statements (1) and (2), p could be 84. However, 84 is not divisible by 168: $84=2\times 2\times 3\times 7$,so you are missing a needed 2.

Both statements mention that p contains at least one 2 in its prime factorization. It is possible that these statements are referring to the same 2. Therefore, one of the 2's in statement (2) overlaps with the 2 from statement (1). You have been given redundant information. The two boxes you made for statements (1) and (2) are not truly different boxes. Rather,they are two different views of the same box (the prime box of p).

Thus, you have to eliminate the redundant 2 when you combine the two views of p's prime box from statements (1) and (2). Given both statements, you only know that p has two 2's, one 3, and one 7 in its prime box.