What is the remainder when the positive integer n is divided by 12? When n is…

Concept:

In Data Sufficiency, a statement is sufficient only when it forces exactly ONE value of the quantity asked — here, the remainder when n is divided by 12 — for every n consistent with it. Test each statement alone first; combine them only if neither alone suffices.

Application:

1. Statement (1): n divided by 6 leaves remainder 1, so n = 6k + 1. Checking k = 0, 1, 2, 3 gives n = 1, 7, 13, 19, whose remainders on division by 12 are 1, 7, 1, 7 — the remainder alternates between 1 and 7, so Statement (1) alone is not sufficient.

2. Statement (2): the remainder on division by 12 exceeds 5, i.e. it is 6, 7, 8, 9, 10, or 11 — many values of n satisfy this, so Statement (2) alone is not sufficient.

3. Combining both: from Statement (1) the remainder is 1 or 7; Statement (2) rules out 1 (since 1 is not greater than 5), leaving only 7. So together the remainder is uniquely 7.

Cross-check:

Check with n = 7 (k = 1, odd): 7 divided by 6 leaves remainder 1 (satisfies Statement 1); 7 divided by 12 leaves remainder 7, which is greater than 5 (satisfies Statement 2); remainder is 7, as expected. Check with n = 1 (k = 0, even): remainder on division by 12 is 1, which is NOT greater than 5, so Statement (2) correctly excludes this case. This confirms the two statements together — but not either alone — fix the remainder at 7.

Answer:

Statements (1) and (2) TOGETHER are sufficient, but neither is sufficient independently (remainder = 7).