If x is a positive integer, what is the least common multiple of x, 6

1. I guess I did it recently.

1) The least common multiple of x and 6 is 30: x could be 5, 10, 15 or 30. nsf.
2) The least common mutiple of x and 9 is 45: x is 5, 15, or 45. nsf.

togather x = 5 and 15 suffff....

C.

GMAT TIGER

Hum......Do you want to give another try?

1) the LCM of x and 6 is 30. That means you need to find the LCM of 30 and 9 --> sufficient

2) LCM of x and 9 is 45 . you need to find out the LCM of 45 and 9 . - > sufficient .

Answer D

First, find LCM of 9 and 6 = 18, and then find LCM of:

(I) 18 and 30 = 90, Suff
(II) 18 and 45= 90, Suff

Answer D

statement (1):
the least common multiple of x and (2)(3) is (2)(3)(5).
this means:
we KNOW x contains exactly one 5, because otherwise that 5 wouldn't be in the lcm.
x MAY contain a 2, a 3, both, or neither; any of these possibilities would yield the (2)(3) in the lcm. note that x cannot contain more than one 2 or 3, as those powers would then go into the lcm.
x cannot contain any other primes, because those primes would have to appear in the lcm, and they don't.
given these facts, you know that the lcm of x, (3)(2), and (3^2) is (2)(3^2)(5). if you don't see why right away, run through the possibilities.

--

statement (2):
use the same kind of analysis as that used for statement (1).
the lcm of x and (3^2) is (3^2)(5).
therefore,
we KNOW that x contains exactly one 5.
x MAY contain no, one, or two 3's.
x CANNOT contain any other primes.
same sort of reasoning used above --> the lcm must be (2)(3^2)(5); sufficient

Hence D.

hi

LCM of x, 6, 9 is the same as the LCM of (x, 6) and 9..
statement 1 provides the LCM of (x, 6) sufficient ....

in the same line of reasoning, LCM of x, 6, 9 is the same as the LCM of ( x, 9) and 6...
statement 2 provides the LCM of (x, 9) sufficient...
D

cheers, if this helps ...

We are given that x is a positive integer and need to determine the LCM of x, 6, and 9. If we can determine the value of x, then we can determine the LCM of x, 6, and 9.

Statement One Alone:

The least common multiple of x and 6 is 30.

Using the information in statement one, we see that x can be 5 since LCM(5, 6) = 30. However, x can be also 10, 15, or 30; any of these values of x would also make LCM(x, 6) = 30. This may seem insufficient to answer the question, but let’s determine the LCM of x, 6 and 9 using these values:

If x = 5, LCM(5, 6, 9) = 90.

If x = 10, LCM(10, 6, 9) = 90.

If x = 15, LCM(15, 6, 9) = 90.

If x = 30, LCM(30, 6, 9) = 90.

Since the LCM of x, 6, and 9 is always 90 for all possible values of x, we see that statement one is sufficient to answer the question.

Statement Two Alone:

The least common multiple of x and 9 is 45.

Using the information in statement two, we see that x can be 5 since LCM(5, 9) = 45. However, x can be also 15 or 45; any of these values of x would also make LCM(x, 9) = 45. This may seem insufficient to answer the question, but let’s determine the LCM of x, 6 and 9 using these values:

If x = 5, LCM(5, 6, 9) = 90.

If x = 15, LCM(15, 6, 9) = 90.

If x = 45, LCM(45, 6, 9) = 90.

Since the LCM of x, 6, and 9 is always 90 for all possible values of x, statement two alone is sufficient to answer the question.

Note: Using either statement, we’ve not determined a unique value for x. However, regardless what the value x is, we’ve determined a unique value for the LCM of x, 6 and 9.

Answer: D

Just use basic concept

A. LCM(x,6,9) = LCM(30,9)=90
B. LCM(x,6,9) = LCM(45,6)=90

IMO D

Target question: What is the LCM of x, 6 and 9?

I'll demonstrate two different approaches.

This first approach uses requires us to be able to think of pairs of values that have given LCM's.
This is a useful skill to have on the GMAT.

Statement 1: The least common multiple of x and 6 is 30.
So, what are some possible values of x?
If the LCM of x and 6 is 30, then x could equal 5, 10, 15 or 30
Let's check each possible value of x.
- If x = 5, then the LCM of x, 6, and 9 is 90
- If x = 10, then the LCM of x, 6, and 9 is 90
- If x = 15, then the LCM of x, 6, and 9 is 90
- If x = 30, then the LCM of x, 6, and 9 is 90
So, even though x can have several different values, it must be the case that the LCM of x, 6, and 9 is 90
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: The least common multiple of x and 9 is 45.
So, what are some possible values of x?
If the LCM of x and 6 is 30, then x could equal 5, 15 or 45
Let's check each possible value of x.
- If x = 5, then the LCM of x, 6, and 9 is 90
- If x = 15, then the LCM of x, 6, and 9 is 90
- If x = 45, then the LCM of x, 6, and 9 is 90
So, even though x can have several different values, it must be the case that the LCM of x, 6, and 9 is 90
Since we can answer the target question with certainty, statement 2 is SUFFICIENT

Answer:D

Cheers,
Brent

Approach #2:

Target question: What is the LCM of x, 6 and 9?

ASIDE: The LCM tells us about the prime factors that numbers have in common.
For example: The LCM of 20 and 12 is 60
60 = (2)(2)(3)(5). So, the prime factorization of 60 has two 2's, one 3, and one 5.

Now examine the prime factorizations of 20 and 12
20 = (2)(2)(5)
12 = (2)(2)(3)
Notice that each prime factorization has no more than two 2's, one 3, and one 5 in it.
Also notice that the combined prime factorizations of 20 and 12 account for the two 2's, one 3, and one 5 that we find in the prime factorization of 60.

Statement 1: The least common multiple of x and 6 is 30
30 = (2)(3)(5)
6 = (2)(3), so we've already accounted for the one 2 and one 3 in the prime factorization of 30
We're missing only a 5
So, the prime factorization of x must have a 5 in it.
The prime factorization of x could also have a 2 or 3 in it, but they aren't required.
So, the possible values of x are 5, 10 (aka 5 times 2), 15 (aka 5 times 3) and 30 (aka 5 times 2 times 3)
As we saw in my earlier post, if we consider all of these possible values of x, the LCM of x, 6 and 9 is always 90
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: The least common multiple of x and 9 is 45.
45 = (3)(3)(5)
9 = (3)(3), so we've already accounted for the two 3's in the prime factorization of 45
We're missing only a 5
Using the same logic as above, the possible values of x are 5, 15 and 45
If we consider all of these possible values of x, the LCM of x, 6 and 9 is always 90
Since we can answer the target question with certainty, statement 2 is SUFFICIENT

Answer: D

Cheers,
Brent

Hi BrentGMATPrepNow, for St.2, to clarify x could also equeal 10,30 right? as LCM of (10,6,9) = 90 too and same with 30.
Thanks Brent

Statement 2: The least common multiple of x and 9 is 45.
For this statement, the only possible values of x are: 5, 15, and 45

Note: The LCM of 10 and 9 is 90 (not 45).
The LCM of 30 and 9 is 90 (not 45)

Get it, thanks BrentGMATPrepNow

Given that x is a positive integer and we need to find what is the least common multiple of x, 6 and 9

LCM(x,6,9) = LCM( LCM(x,6),9 ) = LCM( LCM(x,9),6 )

STAT 1: The least common multiple of x and 6 is 30.
=> LCM(x,6,9) = LCM(LCM(x,6),9) = LCM(30,9) = 3 * 10 * 3 = 90
=> SUFFICIENT

STAT 2: The least common multiple of x and 9 is 45.
=> LCM(x,6,9) = LCM(LCM(x,9),6) = LCM(45,6) = 3*15*2 = 90
=> SUFFICIENT

So, Answer will be D
Hope it helps!

Watch the following video to Learn the Basics of LCM and GCD

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