December 8, 2022

Can a batch of identical cookies be divided evenly between Laurel and Jean without leftovers and without breaking a cookie?

(1) If the batch of cookies were divided among Laurel, Jean, and Marc, there would be one cookie left over.

(2) If Peter eats three of the cookies first, there will be no leftovers when the rest of the cookies are divided evenly between Laurel and Jean.

(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. (This is correct!)

(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.

The question asks whether a batch of cookies can be divided evenly between two people without leftovers. This is a question about divisibility and remainders. For example, if there are 4 cookies, then they can be evenly divided between two people, but if there are 5 cookies, then they cannot be evenly divided between two people.

This Yes/No question can be rephrased as follows: Is the number of cookies even? If so, then Yes, they can be divided evenly, which is sufficient to answer the question. If not, then No, they cannot be divided evenly, which is also sufficient to answer the question (a definitive No is sufficient, just as a definitive Yes is sufficient). If you can’t tell whether the number is odd or even, then you also can’t tell whether the cookies can be divided evenly between two people.

(1) INSUFFICIENT: Test some cases to see what’s possible. You’re only allowed to try values for which statement (1) is true.
Case 1: Laurel, Jean, and Marc each get one cookie, and one is left over, for a total of four cookies. In this case, Yes, the total number of cookies is even.
Case 2: Laurel, Jean, and Marc each get two cookies, and one is left over, for a total of seven cookies. In this case, No, the total number of cookies is not even.
Since the number of cookies could be even or odd, this statement is not sufficient to answer the question.

(2) SUFFICIENT: Test some cases to see what’s possible. You’re only allowed to try values for which statement (2) is true.
Case 1: Laurel and Jean could start with two cookies, which divide evenly; if Peter had already eaten three, then they began with five cookies total. In this case, No, the total number of cookies is not even.
Case 2: Laurel and Jean could start with four cookies, which divide evenly; if Peter had already eaten three, then they began with seven cookies total. In this case, No, the total number of cookies is not even.

Hmm. Is there a way to come up with a Yes answer to the question? If the remaining cookies can be divided evenly between Laurel and Jean, then the number of remaining cookies must be even. If you take that even value and add the three cookies that Peter ate, then you’ll always end up with an odd number of cookies in the starting batch, since any even number plus any odd number always results in an odd number. In other words, there must be an odd number of cookies to start, and so the answer to the question is Always No, the total number of cookies in the batch is not even.

The correct answer is (B): Statement (2) by itself is sufficient to answer the question.