If 'a' and 'b' are positive integers, and 'a + b' is divisible by 7, is 'a' divisible by 7? (Data Sufficiency) (1) 'b' is divisible by 7. (2) 'b' is not divisible by 7.
- ABOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
- BEACH statement ALONE is sufficient.
- CStatement (2) ALONE is sufficient, but statement (1) ALONE is not sufficient.
- DStatement (1) ALONE is sufficient, but statement (2) ALONE is not sufficient.
Show answer & explanationAnswer & explanation
Correct answer: D. Statement (1) ALONE is sufficient, but statement (2) ALONE is not sufficient.
Given that a + b is divisible by 7, we can write a + b = 7k for some integer k. If b is divisible by 7 (Statement 1), then b = 7m for some integer m. Substituting, a + 7m = 7k, which means a = 7k - 7m = 7(k-m). Thus, 'a' is divisible by 7. Statement (1) is sufficient. If b is not divisible by 7 (Statement 2), 'a' could be divisible by 7 (e.g., a=7, b=7, a+b=14; or a=1, b=6, a+b=7, but 'a' is not divisible by 7). Therefore, Statement (2) is not sufficient.
Why the other options are wrong
- A. Statement (1) alone is sufficient.
- B. Only statement (1) is sufficient.
- C. Statement (2) is not sufficient.
Divisibility Rules (Sums/Differences) in DS
Applying properties of divisibility to sums and differences of integers, especially in Data Sufficiency questions.
- If a + b is divisible by n, and 'a' is divisible by n, then 'b' must be divisible by n.
- If a + b is divisible by n, and 'a' is NOT divisible by n, then 'b' must NOT be divisible by n.
- Used to determine if a variable is divisible by a number based on other parts of an expression.
Memory trick: If the sum is divisible, then the parts must follow suit or cancel out.