If x, y, and z are all positive integers, what is the least common mul

Statement 1
LCM of xy = 15, we don't know the value of z
insufficient

Statement 2
The least common multiple of x and z is 18, we don't know the value of z
insufficient

Combining statement 1 and 2
x can be either 1 or 3
1. x=1, y=15 and z=18, LCM = 90
2. x=3, y=5 and z=18, LCM=90
3. x=3, y=15 and z=18 LCM=90
Sufficient

Thanks.. For all value of x & y having LCM as 15, z=18 will always give LCM of x,y,z as 90.
So, we don't need to worry about xz being 1x18 or 9x2 or 6x3... right ?

Exactly!! BTW you can't consider 6*3 because their LCM is 6 not 18

Thanks.. For all value of x & y having LCM as 15, z=18 will always give LCM of x,y,z as 90.
So, we don't need to worry about xz being 1x18 or 9x2 or 6x3... right ?[/quote]

Hi, Bunuel

I want to clear my concept on LCM; Suppose LCM of any set 'n' & set 'm' are given and if we are asked to find LCM of all the no.s of set n & m combined it will simply be LCM [LCM (set n), LCM (set m)] why? as LCM is the least such no. which is divisible by all the no.s of set n & m.

If i'm wrong please clear my doubt!!

Veritas Official Explanation:

This abstract problem rewards students who have a good grasp on what it means for a number to be a least common multiple of two integers and who can easily use prime factorization to find least common multiples. Remember that the least common multiple of two integers can be constructed by first finding their prime factorizations and then combining those prime factorizations, using only the highest exponent to which each prime factor is raised.

Statement (1) gives that the least common multiple of x and y is 15. That means that their combined prime factorization is (3)(5). This does not tell you any information about the values of x and y, but (more importantly) it also gives no information about their relationship to z. You can therefore conclude that statement (1) is insufficient.

Statement (2) gives that the least common multiple of x and z is 18. This means that the combined prime factorization is (2)(32). Again, however, this gives no information about the relationship between x and z and the third variable, y, so statement (2) is insufficient.

Taken together, you know the relationship between x and y and the relationship between x and z. While this gives you no information about the actual values of any variable, the question is about the least common multiple, which only requires that you know 1) the prime factors of each number and 2) that for each prime factor you take the highest exponent for each. You should therefore recognize that the least common multiple of x,y and z can be made by combining the prime factorizations to get a least common multiple of (2)(\(3^2\))(5). Taken together, the two statements are sufficient and choice (C) is correct.

can anybody explain to me why lcm of y and z is not required here and why we have answer as C though we have lcms of xy and xz.

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