If x is a positive integer greater than 1, and y is the smallest posit

Now, had the number y been product of all numbers from 1 to x, y would have been x!. But we are looking at LCM(1 to x) = y.

(1) y = 10x
\(y=10x=1*2*5*x\), so x has to be greater than 5, and should be a multiple of 3 and 4 for sure.
we can test x for the least value after 5, that is 6..
y=10*6=60. CHECK if 60 is divisible by all numbers from 1 to 6 => Answer is YES.
so x=6

(2) y = 60
y is divisible by consecutive numbers 1, 2, 3, 4, 5, 6
Thus x can take any value - 5 or 6.
When x =5, the LCM of (1,2,3,4,5)=60
When x=6, the LCM of (1,2,3,4,5,6)=60


A
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From the first statement, we get y=60 & second statement says y=60. If y is 60 and is the smallest no. divisible between 1 and X, so X has to be six.
How can X take any value from 1 to 2,3,4,5,6? Y is supposed to be the smallest no. so if X is 3 y=2 & if X is 4 Y=6.
Shouldn't the answer be D i.e. from both the statement we can get the value of X?

When we look at the second statement, we can get x as 5 or 6. I have added additional info in my solution above. May clear the query now.
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Hi,

the question tells that y is the smallest integer that has 1 to x as factors. Having factors from 1 to x means that the prime factorization of y can create any value between 1 and x, inclusive of both, 1 and x. For example, if x is 4, then y=(2^2)(3).

(1) y=10x. We try out 6 because (2) is y=60 and it looks like a good bet. We get y=60, and x=6. For x=6, the y as required per question is y=(2^2)(3)(5). So we directly see that it works for x=6. Trying out x=7, we see that the required y is y=(2^2)(3)(5)(7) != 70, and for any x greater than 6, the pattern will continue, meaning that we outgrow our required y. We can do the same for x<6 and we see that only x=6 satifies, hence (1) is suff

(2) If y=60, then the required y is y=(2^2)(3)(5). But here, we can x=5 AND x=6, the latter we have shown in (1). Here is why x=5 is also a valid case. The required y for x=5 is y=(2^2)(3)(5), but that is the same as for 6! So not suff

Hi KarishmaB . Can you please help? There aren't enough solutions here. The ones that are here, are extremely confusing to me.

It is a question based on your understanding of prime factors and LCM

x > 1,
y is the LCM of 1, 2, 3, ... x

Let's take statement 2 first.

(2) y = 60
y = 2*2*3*5
Since y is the LCM of all positive integers 1, 2, 3, ... x, this means 5 must be in this list 1, 2, 3, ... x.
So x is at least 5 (Note that LCM of 1, 2, 3, 4 will not have 5 as a factor. It will have only 2s and 3 as factors)
If x = 5, y will be 60 because 60 is the LCM of 1, 2, 3, 4, 5
If x = 6, y will be 60 because LCM of 1, 2, 3, 4, 5, 6 is again 60.
We don't know whether x is 5 or 6.
Not sufficient alone.

(1) y = 10x
y = 2*5*x
Since y is the LCM of all positive integers 1, 2, 3, ... x, this means 5 must be in this list 1, 2, 3, ... x.
So x must be at least 5. Then y must have 3 as a factor. Since y is 2*5*x, the 3 has to come from x. So x must be a multiple of 3.
The smallest such value of x can be 6. Then y is 2*5*6 = 60 which is also the LCM of 1, 2, 3, 4, 5, 6. This works.
Note that x can take no higher value such as 9 because then y = 2*5*9 which will need to be the LCM of 1, 2, 3, 4, 5, 6, 7, 8, 9 (which it is not since there is no 7 in y and only one 2)
As value of x increases, number of prime factors required increases substantially.
Hence x must be 6.
Sufficient alone.

Answer (A)
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Thanks! KarishmaB

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gmatophobia

What does the question mean by saying that "y is the smallest positive integer that is evenly divisible by every integer" Just that it is LCM?

That's correct. y is the LCM of all integers between 1 and x, inclusive.
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gmatophobia

How can we say that y is the LCM of ALL integers from 1 to x (inclusive)?

The question says that y is evenly divisible with all the integers BETWEEN 1 and x. This means that y is the LCM of integers 2,....,x-1.

chetan2u GMATNinja

rgoel2305 This information is provided in the premise of the question-

If x is a positive integer greater than 1, and y is the smallest positive integer that is evenly divisible by every integer between 1 and x, inclusive, what is the value of x ?

Let me know if you still have any questions.
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Oh. My apologies. I missed that information. Thank you so much.