Question

If 'a' and 'b' are integers, is \( \left( \dfrac{a}{6} + \dfrac{b}{5} \right) \) an integer?

Statement 1: 'a' is divisible by 5 and 'b' is divisible by 6
Statement 2: 'a' is a multiple of 6 which is one-tenth the value of 'b'

• A. If the data in statement (1) alone is sufficient to answer the question
• B. If the data in statement (2) alone is sufficient to answer the question
• C. If the data in both the statements together are needed to answer the question
• D. If either statement (1) alone or statement (2) alone is sufficient to answer the question

Hint:

Substitute statement 2's relation \( b = 10a \) into the expression and simplify before checking statement 1 with numbers.

Answer 1

The Correct Option is B

Step 1: Restate the condition.
The expression \( \dfrac{a}{6} + \dfrac{b}{5} \) equals \( \dfrac{5a + 6b}{30} \).
This is an integer only when \( 5a + 6b \) is a multiple of 30.

Step 2: Test statement 1 alone.
Statement 1 says 'a' is divisible by 5 and 'b' is divisible by 6.
Take a = 5 and b = 6. Then \( 5a + 6b = 25 + 36 = 61 \), which is not a multiple of 30.
Take a = 30 and b = 30. Then \( 5a + 6b = 150 + 180 = 330 \), which is a multiple of 30.
Two valid cases give different results, so statement 1 alone does not settle the question.

Step 3: Test statement 2 alone.
Statement 2 says 'a' is a multiple of 6 and b = 10a.
Write a = 6n for some integer n, so b = 60n.
Then \( \dfrac{a}{6} + \dfrac{b}{5} = n + 12n = 13n \), which is always an integer for any integer n.
This holds for every valid value of n, so statement 2 alone always gives a definite yes.

Final Answer:
Statement 2 alone is sufficient, while statement 1 alone is not. \[ \boxed{(b)} \]