If a and b are positive integers, is ab even?
1. a^b is odd
2. a + b is even
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.
Recall that the product of two integers is even if one or both of the integers are even. Therefore, ab is even if a or b is even. Statement 1 tells us that a^b is odd. Remember that the product of two or more integers is odd only if all of the integers are odd. Since a^b is equal to a × a × ...
× a (b times), we know that a must be odd. However, a^b is odd whether the exponent, b, is even or odd. Therefore, we do not know whether b is even or odd, and so we do not know whether a^b is even.
Statement 1 is NOT sufficient.
Statement 2 tells us that the sum a + b is even. Remember that the sum of two integers is even if both integers are odd or if both integers are even. Therefore, if a is odd, then b is also odd. Alternatively, if a is even, b is also even.
Yet if a and b are odd, then ab is odd; if a and b are even, then ab is even. Therefore, the product of ab could be either odd or even. Statement 2 is NOT sufficient.
Now combine statements 1 and 2. We know from statement 1 that a is odd. If a is odd, then a + b is even only if b is odd. Therefore, we know that a and b are both odd and that ab is, therefore, odd.
The two statements together are sufficient.
Answer - Choice C
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