danzig wrote:
In the fraction x/y, where x and y are positive integers, what is the value of y ?
(1) x is an even multiple of y.
(2) x - y = 2
I imagine most people will solve here by testing numbers, since it's easy enough to get a feeling for what's happening by doing that. There is a theoretical justification for the answer here that does not involve number picking:
The answer is clearly C or E. Using Statement 2, we know x and y are 2 apart. Many test takers learn that when two integers are 1 apart (i.e. when integers are consecutive), then their GCD is 1. We can extend that fact. Say two numbers are 2 apart, so our numbers are y and y + 2. Imagine y is a multiple of 3. If y is a multiple of 3, the next larger multiple of 3 will be y + 3, not y + 2. Similarly if y is a multiple of k, where k > 2, then the next multiple of k will be y + k, not y + 2. So y and y + 2 can't possibly share any divisor larger than 2, and their GCD is either 1 (if they are both odd) or 2 (if they are both even). In general, the GCD of two numbers y and y + m, i.e. two numbers that are m apart, must always be a factor of m, so for example, the GCD of c and c + 21 can only ever be 1, 3, 7 or 21.
So here, when we use both Statements, we know both our numbers are even (since x is, and x and y are 2 apart). We know their GCD can only be 1 or 2 from Statement 2, and if both numbers are even, their GCD must be 2. But we also know x is a multiple of y. If that's true, the GCD of x and y is automatically equal to y (since y is clearly then a divisor of both x and y, but no larger number is a divisor of y). If the GCD of x and y is equal both to 2 and to y, then y = 2. So the answer is C.
That said, I find it mystifying why someone would word a question in this way. What does "the fraction x/y" have to do with anything? Why doesn't the question just ask "If x and y are positive integers, what is the value of y"? The wording is copied from an old official question, but in the official question, the fraction was crucially important (the question went on to talk about common denominators), and the fraction is irrelevant here.
Statement 1 is also potentially ambiguous, and if this were an official problem, I'm almost certain they would have chosen a non-ambiguous wording. I expect Statement 1 is meant to be interpreted to mean "x is even and x is a multiple of y". But I don't think it's completely unreasonable to interpret Statement 1, as written, to mean "x is equal to an even number times y", which is not the same thing. And colloquially, sometimes people say one number is "evenly divisible" by another, and when people say that, the meaning of "even" has nothing to do with multiples of 2 -- it means there is no remainder. So here, I also think it's not completely unreasonable to interpret the phrase "even multiple" to mean "a multiple with no remainder". The Statement really should be worded differently so the test taker isn't left to guess what it intends.
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