If A and B are integers, is B even?

Step 1: Analyse Question Stem

A and B are integers.
We have to find if B is even.

Step 2: Analyse Statements Independently (And eliminate options) – AD / BCE

Statement 1: 14A – 11B = 7N

There is no information about N; it could be a fraction or an integer.
The best strategy is to reorganise the equation to have B on one side and then test cases.

Reorganising, 11B = 14 A – 7 N.

If B = 1, A = 1 and N = \(\frac{3}{7}\), the equation is satisfied. In this case, B is not even.

If B = 2, A = 2 and N = \(\frac{6}{7}\), the equation is satisfied. In this case, B is even.

The data in statement 1 is insufficient to answer the question with a definite YES or NO.
Statement 1 alone is insufficient. Answer options A and D can be eliminated.

Statement 2: 14A – 11B = 2N

Again, there is no information about N. We can follow a similar strategy of reorganising the equation and testing cases.

Reorganising, 11B = 14 A – 2N

If B = 1, A = 1 and N = \(\frac{3}{2}\), the equation is satisfied. In this case, B is not even.

If B = 2, A = 2 and N = 3, the equation is satisfied. In this case, B is even.

The data in statement 2 is insufficient to answer the question with a definite YES or NO.
Statement 2 alone is insufficient. Answer option B can be eliminated.

Step 3: Analyse Statements by combining

From statement 1: 14A – 11B = 7N
From statement 2: 14A – 11B = 2N

Therefore, 7N = 2N. This is possible only if N = 0.

Therefore, 14 A – 11 B = 0; or 14 A = 11 B.

Hence, \(\frac{A }{ B}\) = \(\frac{11 }{ 14}\).

Since B is an integer, it can be concluded that B is a multiple of 14. Any multiple of 14 is always even.
Therefore, B is even.

The combination of statements is sufficient to answer the question with a definite YES.
Statements 1 and 2 together are sufficient. Answer option E can be eliminated.

The correct answer option is C.