Question

Is \( n \) odd?
[I.] \( n \) is divisible by 3, 5, 7, and 9.
[II.] \( 0 < n < 400 \)

• A. if the question can be answered with the help of any one statement alone but not by the other statement.
• B. if the question can be answered with the help of either of the statements taken individually.
• C. if the question can be answered with the help of both statements together.
• D. if the question cannot be answered even with the help of both statements together.

Hint:

Check LCM and apply the range to narrow down to a unique value.

Answer 1

The Correct Option is C

From statement I: If \( n \) is divisible by 3, 5, 7, and 9, then it must be a multiple of their LCM. \[ \text{LCM}(3,5,7,9) = \text{LCM}(3,3^2,5,7) = 3^2 \times 5 \times 7 = 315 \] So, \( n = 315k \) for some integer \( k \). From statement II: \( 0 < n < 400 \) Combining both: The only multiple of 315 in that range is \( n = 315 \), which is an odd number. So, both statements together are required to determine that \( n \) is odd.