Question:medium

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'

Show Hint

Substitute statement 2's relation \( b = 10a \) into the expression and simplify before checking statement 1 with numbers.
Updated On: Jul 21, 2026
  • If the data in statement (1) alone is sufficient to answer the question
  • If the data in statement (2) alone is sufficient to answer the question
  • If the data in both the statements together are needed to answer the question
  • If either statement (1) alone or statement (2) alone is sufficient to answer the question
Show Solution

The Correct Option is B

Solution and Explanation

Step 1: Rewrite the target using a common denominator.
$ \dfrac{a}{6} + \dfrac{b}{5} = \dfrac{5a + 6b}{30} $, so the real question is whether 30 divides \( 5a + 6b \).

Step 2: Check statement 2 first.
Statement 2 fixes b = 10a with a itself a multiple of 6.
Substitute directly: \( 5a + 6b = 5a + 60a = 65a \).
Since a is a multiple of 6, write a = 6n, giving \( 65a = 390n = 30 \times 13n \).
This is always a multiple of 30, so the sum is always an integer. Statement 2 alone answers the question with a firm yes.

Step 3: Check statement 1 next.
Statement 1 only fixes a as a multiple of 5 and b as a multiple of 6, without linking them.
With a = 5 and b = 6, the sum works out to \( 61/30 \), not an integer.
With a = 30 and b = 30, the sum works out to \( 330/30 = 11 \), an integer.
Because the outcome changes across valid cases, statement 1 alone cannot give one fixed answer.

Final Answer:
Only statement 2 pins down a single answer by itself. \[ \boxed{(b)} \]
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