Question

If 'a' and 'b' are integers, is \((\frac{a}{6}+\frac{b}{5})\) an integer?
Statement 1: 'a' is divisible by 5 and 'b' is divisible by 6
Statement 2: 'a' is multiple of 6 which is one-tenth the value of 'b'

• A. Statement (1) alone is sufficient to answer the question
• B. Statement (2) alone is sufficient to answer the question
• C. Both the statements together are needed to answer the question
• D. Either statement (1) alone or statement (2) alone is sufficient to answer the question
• E. Neither statement (1) nor statement (2) suffices to answer the question.

Answer 1

The Correct Option is B

To determine if \(\left( \frac{a}{6} + \frac{b}{5} \right)\) is an integer, we need information about the divisibility and relationships between the integers \(a\) and \(b\). Let's analyze each statement:

Statement 1:

"\(a\) is divisible by 5 and \(b\) is divisible by 6." 

• Since \(a\) is divisible by 5, we can express \(a\) as \(a = 5k\) for some integer \(k\).
• Since \(b\) is divisible by 6, we can express \(b\) as \(b = 6m\) for some integer \(m\).
• Substituting these values into the expression: \(\frac{a}{6} + \frac{b}{5} = \frac{5k}{6} + \frac{6m}{5} = \frac{25k + 36m}{30}\).
• For the expression \(\frac{25k + 36m}{30}\) to be an integer, \(25k + 36m\) must be divisible by 30. This is not always guaranteed by just knowing the individual divisibility, as 25 and 36 do not individually cover all factors of 30.

Statement 1 alone is not sufficient.

Statement 2:

"\(a\) is a multiple of 6 which is one-tenth the value of \(b\)."

• This implies that \(a = 6n\) for some integer \(n\) and \(b = 10a = 60n\).
• Substitute these in the expression: \(\frac{a}{6} + \frac{b}{5} = \frac{6n}{6} + \frac{60n}{5} = n + 12n = 13n\).
• Here, \(13n\) is clearly an integer as \(n\) is an integer.

Statement 2 alone is sufficient.

Conclusion:

Based on the analysis, Statement (2) alone is sufficient to answer the question, resulting in the conclusion that \((\frac{a}{6}+\frac{b}{5})\) is indeed an integer. Therefore, the correct answer is "Statement (2) alone is sufficient to answer the question."