MGMAT Challenge Problems
08/12/02
DS Question
sqrt ABC=504
Is B divisible by 2?
(1) C = 168
(2) A is a perfect square
(A) Statement (1) alone is sufficient, but statement (2) alone is not sufficient.
(B) Statement (2) alone is sufficient, but statement (1) alone is not sufficient.
(C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
(D) Each statement ALONE is sufficient.
(E) Statements (1) and (2) TOGETHER are NOT sufficient.
Solution
Divisibility problems can be solved using prime factorization.
The prime factorization of 504=2^3*3^2*7
Therefore, using the given equation, we can see that:
sqrt ABC=504
ABC=2^6* 3^4*7
To answer the question, we must determine whether B contains one of the six 2's in the prime factorization.
Statement (1) alone tells us that 168=2^3*3*7
This tells us that C has three of the 2's in the prime factorization. However, since we have no information about A or B, this is not sufficient information to answer the question.
Statement (2) alone tells us that A is a perfect square.
This tells us that if A contains any 2's as prime factors, it must have an even number of 2's. (The only way a number can be a perfect square is if its prime factors come in pairs). This, again is not sufficient information to answer the question.
Using both statements together, we know that C has three of the 2's. We also know that A can have either zero of the 2's or two of the 2's, but, since A is a perfect square, it cannot have all three of the remaining 2's.
Thus, B must have at least one 2 as a prime factor. The correct answer is C.
Why answer is C not E
What about if A is 16 or 25 then B becomes odd number