If x and y are positive integers such that 1

Question

If x and y are positive integers such that 1< x < y, is y divisible by x?

(1) x and y have the same unique prime factors

(2) x and y have the same number of prime factors

IanStewart · Answer

The mathematics in this question is absolutely wrong if the OA is "B" -- what is the source?

Using both Statements, it might be that x = 6 and y = 12, and then y is divisible by x, or it might be that x = 12 and y = 18, and then y is not divisible by x. In each case, x and y have the same prime factors, 2 and 3, and x and y both have two prime factors, so have the same number of prime factors.

Russ19 · Answer

Hi

The problem above is from TTP, and I am going to attach a file here.

Thanks

mSKR · Answer

1< x < y

(1) x and y have the same unique prime factors(2,3,5)
x=2,3,3,5=90
y= 2,3,5,5=150

y not divisible by X

x=2,3,5=30
y= 2,3,5,5=150
y divisible by X

not sufficient

2.)

x=2,2,2,2
y= 3,3,3,3
y>x and same number of PRIME factors
not divisible

Y is always greater than x
prime factoes would be different

and hence Y can never be divisible by X

Hence B

chetan2u · Answer

So, y should be a multiple of x..

(1) x and y have the same unique prime factors
 x=2^3*5^1, y=2^3*5^2 ...YES
 x=2^3*5^1, y=2^1*5^2 ...NO

(2) x and y have the same number of prime factors
Although such language is rarely seen on GMAT, but I assume the statement means that the number of prime factors means repetition is allowed as in 4=2^2, so two prime factors 2 and 2, or 2^2*3^3 will have 5 prime factors -2,2,3,3,3)
If x and y have the same number of prime factors, they can be SAME, that is x=y, in which case answer is YES, but x<y so not possible.
In all other cases, there will always be a prime factor(even if repeated) of x that will not exist in y.
Say  x=a^p*b^q*c^r , then y can be  d^p*b^{q+r} ...does not contain a, so answer is NO
or Say  x=a^p*b^q*c^r , then y can be  a^p*b^{q-1}*c^{r+1} ...does not contain b, so answer is NO
Sufficient

B

IanStewart · Answer

I think you're correctly guessing what the question intends, but the question is wrong. We shouldn't try to find some justification for it, because test takers who think it's right will answer many official questions incorrectly. So we shouldn't say this language is "rarely seen" on the GMAT -- this language is never seen on the GMAT, because it is mathematically incorrect. The number 9, for example, has only one prime factor, 3. It does not have two prime factors; there is just no reason to count the '3' twice. If anyone needs evidence of what the actual GMAT means by phrases like this (and this is also what any Number Theorist in the world means -- GMAT math is the same as real math), you can look at this official question: https://gmatclub.com/forum/the-positive ... 60634.html That question begins with the sentence "The positive integer k has exactly two positive prime factors, 3 and 7". Notice what that means: it means k = (3^a)(7^b), where a and b are positive integers. It does not mean that k = 21, which is the conclusion you'd reach if you thought we were supposed to count prime factors the way the question in this thread claims you're supposed to. If anyone wants to see the kind of wording the GMAT needs to use when you *are* supposed to count prime factors *with* repetitions, you can see this question (the only official problem I've ever seen where you're supposed to do that, incidentally) -- https://gmatclub.com/forum/for-any-posi ... 90320.html Notice just how much additional explanation that question needs to include to be perfectly clear that you are supposed to count repeated primes -- it even gives a numerical example. It does not even use the phrase "number of prime factors", because that would potentially be confusing. edit: I'm not sure why, but when I first wrote this post two years ago, the link to the official question did not appear, probably my mistake, so I've just added it above in red

chetan2u · Answer

IanStewart , I would totally agree with you on your point. Just to add on to your point - We talk of number of factors of a number a^x*b^y given by (x+1)(y+1). Now, say the number is 8=2*2*2, the factors are clearly 1,2,4,8 and not 1,2,2,2,4,4,8. ‘Rarely’ was just to give some respect to the creator even though it is faulty and that is why I said what it is trying to mean and not what it means.

I agree the question is poorly written and would not ever be close to actual.

Braintree · Answer

Thanks, Ian, for such a great explanation. Btw, can you please post the link to the question for which you refer to in your above post as "If anyone wants to see the kind of wording the GMAT needs to use when you *are* supposed to count prime factors *with* repetitions.." ? Somehow the link to that question is not appearing in your post above.

Kind Regards,
BT

IanStewart · Answer

Thanks for pointing that out! I'm not sure what happened there, but I have edited my post to include the correct link.

If x and y are positive integers such that 1< x < y, is y divisible by

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