If 'a' and 'b' are integers, is 'a' divisible by 4? (Data Sufficiency) (1) a + b is divisible by 4. (2) b is divisible by 4.
- AStatement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
- BStatement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
- CEACH statement ALONE is sufficient.
- DBOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
Show answer & explanationAnswer & explanation
Correct answer: D. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
Statement (1) alone: a + b = 4k for some integer k. If b=0, a=4k, so 'a' is divisible by 4. If b=1, a=4k-1, so 'a' is not necessarily divisible by 4 (e.g., a=3, b=1, a+b=4). Not sufficient. Statement (2) alone: b is divisible by 4. This tells us nothing about 'a'. 'a' could be 4 (divisible by 4) or 5 (not divisible by 4). Not sufficient. Combining (1) and (2): We know a + b is divisible by 4, and b is divisible by 4. Let a + b = 4k and b = 4m for some integers k, m. Substitute b into the first equation: a + 4m = 4k. Then a = 4k - 4m = 4(k - m). Since k and m are integers, k - m is an integer. Therefore, 'a' is divisible by 4. Both statements together are sufficient.
Why the other options are wrong
- A. Incorrect. Statement (2) alone is not sufficient (b can be divisible by 4, but 'a' can be anything).
- B. Incorrect. Statement (1) alone is not sufficient (e.g., a=3, b=1).
- C. Incorrect. Neither statement alone is sufficient.
Divisibility Rules (Sums/Differences) in DS
If (A + B) is divisible by N, and B is divisible by N, then A must also be divisible by N. This principle is key for Data Sufficiency problems involving divisibility.
- If a sum/difference of two numbers is divisible by N, and one number is divisible by N, the other must also be divisible by N.
- This extends to remainders: if A = qN + r and B = pN, then A+B = (q+p)N + r, so (A+B) has the same remainder as A when divided by N.
- Crucial for determining divisibility of an unknown variable.
Memory trick: If the sum shares a trait, and one part has it, the other part must too.