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If x is a positive integer, what is the least common

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by VJesus12 » Mon May 14, 2018 1:33 am

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Answers

A

B

C

D

E

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Difficulty—

If x is a positive integer, what is the least common multiple of x, 6 and 9?

(1) The least common multiple of x and 6 is 30.
(2) The least common multiple of x and 9 is 45.

The OA is the option D.

I am really confused here. Could someone give me a good explanation of why is D the correct answer?
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by GMATGuruNY » Mon May 14, 2018 3:02 am
VJesus12 wrote:If x is a positive integer, what is the least common multiple of x, 6 and 9?

(1) The least common multiple of x and 6 is 30.
(2) The least common multiple of x and 9 is 45.
Statement 1:
x must be a factor of 30 such that the smallest value divisible by both x and 6 is 30.
Options for x:
x=5, 10, 15, 30.
In every case, the LCM for x , 6, and 9 is 90.
SUFFICIENT.

Statement 2:
x must be a factor of 45 such that the smallest value divisible by both x and 9 is 45.
Options for x:
x=5, 15, 45.
In every case, the LCM for x , 6, and 9 is 90.
SUFFICIENT.

The correct answer is D.
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by Brent@GMATPrepNow » Mon May 14, 2018 5:44 am
VJesus12 wrote:If x is a positive integer, what is the least common multiple of x, 6 and 9?

(1) The least common multiple of x and 6 is 30.
(2) The least common multiple of x and 9 is 45.
Target question: What is the LCM of x, 6 and 9?

I'll show you 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
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by Brent@GMATPrepNow » Mon May 14, 2018 5:45 am
VJesus12 wrote:If x is a positive integer, what is the least common multiple of x, 6 and 9?

(1) The least common multiple of x and 6 is 30.
(2) The least common multiple of x and 9 is 45.
Another approach:

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
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by Jeff@TargetTestPrep » Wed May 16, 2018 10:13 am
VJesus12 wrote:If x is a positive integer, what is the least common multiple of x, 6 and 9?

(1) The least common multiple of x and 6 is 30.
(2) The least common multiple of x and 9 is 45.
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

Jeffrey Miller
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