If a, b, and c are distinct positive integers, is \((\frac{a}{\frac{b}{c}})\) an integer?

Question: If a, b, and c are distinct positive integers, is \((\frac{a}{\frac{b}{c}})\) an integer?

1. \(\frac{a}{c}=3\)
2. a = b + c

1. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
2. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
3. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
4. EACH statement ALONE is sufficient.
5. Statements (1) and (2) TOGETHER are NOT sufficient

“If a, b, and c are distinct positive integers, is \((\frac{a}{\frac{b}{c}})\) an integer?”- is the topic of GMAT Quantitative reasoning section of GMAT. This question has been taken from the book “GMAT Official Guide 2022”. GMAT Quant section consists of a total of 31 questions. GMAT Data Sufficiency questions consist of a problem statement followed by two factual statements.GMAT data sufficiency comprise 15 questions which are two-fifths of the total 31 GMAT quant questions.

Answer:

Simplify the equation \((\frac{a}{\frac{b}{c}})\)as\(\frac{a}{bc}\).

From statement (1), we get

\(\frac{a}{c}=3\)

Put this value in the above equation, we get:

\(\frac{a}{bc}=\frac{3}{b}\)

Now we can see that if b = 1 or b = 3

Then, \((\frac{a}{\frac{b}{c}})\) is an integer but if b is some other positive integer then \((\frac{a}{\frac{b}{c}})\) is not an integer.

Hence, this is not sufficient.

Option A is not correct

Now, From statement (2), we get

a = b + c

Substitute the value of ‘a’ in equation \(\frac{a}{bc}\)

We will get:

\(\frac{a}{bc}=\frac{b+c}{bc} \)So,\(\frac{a}{bc}=\frac{b+c}{bc}=\frac{1}{c}+\frac{1}{b}\)

Now, for this expression to be an integer

either b = c = 1 must be true

OR

b = c = 2 must be true

But it is given that these unknowns are distinct, so neither of the option is possible.

Hence we can say that \(\frac{1}{c}+\frac{1}{b}\) \(\neq\) integer.

This is sufficient.

Correct option: B

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