One of the most common concepts tested on the GMAT quantitative section is divisibility, and the data sufficiency problem below is a prime example of how the exam can ask you to use your factoring/divisibility skills. Take a shot at this question, post your answers (and show your work!) in the comments field, and check back later for a full solution.
For positive integers x and y, x^2 = 350y. Is y divisible by 28?
(1) x is divisible by 4
(2) x^2 is divisible by 28
UPDATE: Solution!
Solution: A. In order to be divisible by 28, a number must have the prime factors 2*2*7. From the given information we can conclude that x^2 = 2*5*5*7*y (the factors indicate the prime factorization of 350), and that, because we need the factors to come in pairs in order for the square root of x^2 to be an integer, y is then divisible by at least 2 and 7. From that, we need the statements to supply one more factor of 2 for y. Statement 1 does exactly that: because x is divisible by 4, then x^2 will be divisible by 4*4, or 2*2*2*2. As 350 only has one 2, y must supply the rest, so we can conclude that y is divisible by 2*2*2*7. As only 2*2*7 was required for y to be divisible by 28, statement 1 is sufficient. Statement 2 only tells us that y is divisible by 2*7 – and we already knew that. Accordingly, statement 2 is not sufficient, and the correct answer is A.
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I will go with 1 . because .. if X^2 is having 16 only then we are assured that 7X5^2X2 needs to be multiplied by atleast 8 X 7 to make primes factors exponent equal on both sides.
For B) ARGUMENT we can have just 7X2 multiplied to Right side. So not enough
IMO A.
x^2=7*5*5*2*y
x= root7*5*root2*rooty
because x and y are integers:
y needs to have – at least: 7, 2 so X will be integer.
stat 1 tells us that x is dividable by 4 so it must have 2*2 right now we have
root7*root7*root2*root2*5 —> y have to have at least a couple more 2*2 so that root x would stay integer and would be dividable by 4.
so y=7*2*2*2 at least. therefore y will always be dividable by 28 (7*2*2).
stat 2 – x^2 is dividable by 28 is insufficient.
x^2= 7*5*5*2*y —> Y can be 2 (cannot be dividable by 2) or 28.
so IMO A.
hope im not wrong. thanks for this great question.
correction:
x^2= 7*5*5*2*y —> Y can be 4 (cannot be dividable by 2) or 196.
Based on the answer posted [quote]Statement 2 only tells us that y is divisible by 2*7 [/quote]
From the approach I followed I found out that Statement 2 tells us that y is divisible by 2.
Could you please explain how you got that y is divisible by 2*7 from statement 2?
Thanks
Hey Sony,
Sure thing! So statement 2 says:
x^2 is divisible by 28
Which we can rephrase as:
x*x = 2*2*7*(?) (where the question mark could be “anything else” – we know that x^2 is divisible by, but not limited to, 28).
Because there are two x’s, we need to give each x an equal set of factors. The twos evenly divide so that each x gets its own 2. But the 7 does not. However, look at the question stem, which says “for integers x and y”. Sqrt(7) is not an integer, so the only way for x to be an integer is for each x to get its own 7…any other way and we don’t have an integer. So we now know that each x gets a 2 and a 7 (and maybe something else), so at minimum x = 2*7.
perfect! i get it so far.. and then how do we continue our reasoning to conclude that y has only 2 and 7 as factors?
Thanks again!
Brian?? could you please tell me how we continue our reasoning?
Thanks
sorry I meant at least 2 and 7 as factors..
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