Question

If x and y are both positive integers, x is a multiple of 3 and y is a multiple of 21, is xy a multiple of 75? x is a multiple of 9. y is a multiple of 25. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked Both statements (1) and (2) TOGETHER are sufficient to answer the question asked; but NEITHER statement ALONE is sufficient EACH statement ALONE is sufficient to answer the question asked Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed

Answer 1

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked

Explanation:

For xy be a multiple of 75, x and y must be able to be factorized into numbers that multiply 75. A group of prime numbers that multiplied give 75 is {3,3,5} (3 * 5 * 5 = 75).

The question says x is a multiple of 3, this means x=3p and y is a multiple of 21 then y=21q. With this:

xy=3p*21q

xy=3*21*pq

xy=3*3*7*pq

The statement (1) says x is a multiple of 9, to include this information we need another 3 has a factor in x because we already know x is a multiple of 3 (3*3=9), then x=3*3*r. For xy this means

xy=3*3*r*21*q

xy=3*3*3*7*r*q

The group of numbers {3,3,3,7} multiplied can't give 75, maybe r or q have the 5 that are needed but we don't know it.

The statement (2) says y is a multiple of 25 but y is also a multiple of 21, then y=21*25*s, including this for xy (Don't include the information given by the statement (1) to see if the information of statement 2 alone is enough):

xy=3p*21*25s

xy=3*3*7*5*5*r*s

In the group of numbers {3,3,7,5,5}, we have the two 5 and at least one 3 to form 75, with this information is possible to say if xy is a multiple of 75.