In the fraction x/y, where x and y are positive integers, what is the value of y? The least common...

Understanding the Question

We need to find the exact value of y in the fraction \(\mathrm{x/y}\), where both x and y are positive integers.

What We Need to Determine

For sufficiency, we must find exactly one specific value for y. If multiple values are possible, that's not sufficient for a "value" question.

Given Information

• \(\mathrm{x/y}\) is a fraction
• Both x and y are positive integers
• We need the specific value of y

Analyzing Statement 1

Statement 1: The least common denominator of \(\mathrm{x/y}\) and \(\mathrm{1/3}\) is 6.

Let's understand what this means. The LCD of two fractions is the smallest number that both denominators divide into evenly.

Here's the key insight: Since LCD = 6 and we know \(\mathrm{6 = 2 \times 3}\), let's think about what this tells us:

• The fraction \(\mathrm{1/3}\) already has denominator 3
• For the LCD to be 6 (not just 3), the denominator y must contribute the factor 2
• This means y must be divisible by 2

But wait—which multiples of 2 give us LCD = 6?

• If y = 2: \(\mathrm{LCD(2, 3) = 6}\) ✓ (Works!)
• If y = 4: \(\mathrm{LCD(4, 3) = 12 \neq 6}\) (Too large)
• If y = 6: \(\mathrm{LCD(6, 3) = 6}\) ✓ (Also works!)

So y could be either 2 or 6. Since we have two possible values, Statement 1 is NOT sufficient.

[STOP - Not Sufficient!] This eliminates choices A and D.

Analyzing Statement 2

Now let's forget Statement 1 completely and analyze Statement 2 independently.

Statement 2: x = 1

This tells us our fraction becomes \(\mathrm{1/y}\). But knowing that x = 1 tells us nothing about y. Since y can be any positive integer (1, 2, 3, 4, ...), we have infinitely many possibilities.

Statement 2 is NOT sufficient.

[STOP - Not Sufficient!] This eliminates choices B and D (already eliminated).

Combining Statements

Now let's use both statements together:

• From Statement 1: y must be either 2 or 6
• From Statement 2: x = 1

This gives us the fraction \(\mathrm{1/y}\) where y is either 2 or 6. So our fraction could be:

• \(\mathrm{1/2}\), or
• \(\mathrm{1/6}\)

Even with both pieces of information, we still cannot determine which specific value y takes.

The statements together are NOT sufficient.

[STOP - Not Sufficient!] This eliminates choices C (and A, B, D already eliminated).

The Answer: E

Since neither statement alone nor both statements together allow us to determine a unique value for y, the answer is E.

Answer Choice E: "The statements together are not sufficient."