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In the fraction \(\frac{\mathrm{x}}{\mathrm{y}}\), where \(\mathrm{x}\) and \(\mathrm{y}\) are positive integers, what is the value of \(\mathrm{y}\)?
We need to find the exact value of y in the fraction \(\mathrm{x/y}\), where both x and y are positive integers.
For sufficiency, we must find exactly one specific value for y. If multiple values are possible, that's not sufficient for a "value" question.
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:
But wait—which multiples of 2 give us LCD = 6?
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.
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).
Now let's use both statements together:
This gives us the fraction \(\mathrm{1/y}\) where y is either 2 or 6. So our fraction could be:
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).
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."