Question

If k is an integer such that 56 < k < 66, what is the value of k?
(1) If k were divided by 2, the remainder would be 1.
(2) If k + 1 were divided by 3, the remainder would be 0.

• 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.

Answer 1

The correct option is E Statements (1) and (2) together are not sufficient.

We have to determine the value of the integer k, where 56 < k < 66.

1. It is given that the remainder is 1 when k is divided by 2, which implies that k is odd. Therefore, the value of k can be 57, 59, 61, 63, or 65; NOT sufficient.
2. It is given that the remainder is 0 when k + 1 is divided by 3, which implies that k + 1 is divisible by 3. Since 56 < k < 66 (equivalently, 57 < k + 1 < 67), the value of k + 1 can be 60, 63, or 66 so the value of k can be 59, 62 or 65; NOT sufficient.

• Taking (1) and (2) together, 59 and 65 appear in both lists of possible values for k; NOT sufficient.
• The correct answer is E; both statements together are still not sufficient.