Prime Factors problem

[faraz_jeddah]

If x, y, and z are positive integers, what is the greatest prime factor of the product xyz?

(1) The greatest common factor of x, y, and z is 7.

(2) The lowest common multiple of x, y, and z is 84.


Just when I thought I was good with prime factors questions, I encountered this tough cookie.
Although I have the explanation and OA, I feel that this a good question to share with everyone.

[Brent@GMATPrepNow]

If x, y, and z are positive integers, what is the greatest prime factor of the product xyz?

(1) The greatest common factor of x, y, and z is 7.
(2) The lowest common multiple of x, y, and z is 84.

Target question: What is the greatest prime factor of the product xyz?
Let's first clarify what the target question is asking.
It's essentially saying that, if we find the prime factorization of xyz, we want to determine the biggest prime number in this factorization.
Example: 120 = (2)(2)(2)(3)(5). Here, the biggest prime factor is 5

Statement 1: The greatest common factor of x, y, and z is 7
There are several conflicting sets of values that meet this condition. Here are two:
Case a: x = 7, y = 7 and z = 7, in which case xyz = (7)(7)(7), which means the greatest prime factor of the product xyz is 7
Case b: x = 7, y = 7 and z = 77, in which case xyz = (7)(7)(77) = (7)(7)(7)(11), which means the greatest prime factor of the product xyz is 11
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: The lowest common multiple (LCM) of x, y, and z is 84.
This tells us that 84 is a multiple x, 84 is a multiple y, and 84 is a multiple z
Notice that 84 = (2)(2)(3)(7)
If 84 is the LOWEST common multiple (LCM), none of the numbers (x, y or z) can have a number bigger than 7 in their prime factorization.
Also, at least one of the numbers (x, y or z) must have a 7 in its prime factorization (otherwise the LCM would not have a 7 in its prime factorization).
All of this tells us that the prime factorization of xyz includes at least one 7 AND it does not include any primes greater than 7.
So, we can be certain that the greatest prime factor of the product xyz is 7

Answer = B

Cheers,
Brent

[faraz_jeddah]

Thanks Brent.

What threw me off was

(1) The greatest common factor of x,y and Z is 7

[sana.noor]

B for me, LCM means least common multiple which includes all the prime factors of a number. if lcm is given you dont need anything else

[ani781]

1 states => x , y , z have a GCF of 7. So , there is a common prime factor 7 for each of x , y and z. However, each of these can have 11, 17 , 23 etc as a factor , which might not be common to all of x, y & z. So (1) is not sufficient. Eliminate A & D.

2 states => LCM of x,y & z is 84 = 2^2, 3, 7 are the factors . And this set is also the super-set of prime numbers. So there can't be anything greater than 7 as a prime factor. Hence answer is B

[gmattesttaker2]

If x, y, and z are positive integers, what is the greatest prime factor of the product xyz?

(1) The greatest common factor of x, y, and z is 7.
(2) The lowest common multiple of x, y, and z is 84.

Target question: What is the greatest prime factor of the product xyz?
Let's first clarify what the target question is asking.
It's essentially saying that, if we find the prime factorization of xyz, we want to determine the biggest prime number in this factorization.
Example: 120 = (2)(2)(2)(3)(5). Here, the biggest prime factor is 5

Statement 1: The greatest common factor of x, y, and z is 7
There are several conflicting sets of values that meet this condition. Here are two:
Case a: x = 7, y = 7 and z = 7, in which case xyz = (7)(7)(7), which means the greatest prime factor of the product xyz is 7
Case b: x = 7, y = 7 and z = 77, in which case xyz = (7)(7)(77) = (7)(7)(7)(11), which means the greatest prime factor of the product xyz is 11
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: The lowest common multiple (LCM) of x, y, and z is 84.
This tells us that 84 is a multiple x, 84 is a multiple y, and 84 is a multiple z
Notice that 84 = (2)(2)(3)(7)
If 84 is the LOWEST common multiple (LCM), none of the numbers (x, y or z) can have a number bigger than 7 in their prime factorization.
Also, at least one of the numbers (x, y or z) must have a 7 in its prime factorization (otherwise the LCM would not have a 7 in its prime factorization).
All of this tells us that the prime factorization of xyz includes at least one 7 AND it does not include any primes greater than 7.
So, we can be certain that the greatest prime factor of the product xyz is 7

Answer = B

Cheers,
Brent

Hello Brent,

Thanks for your explanation. I just had a question in statement 2:

If 84 is the LOWEST common multiple (LCM), none of the numbers (x, y or z) can have a number bigger than 7 in their prime factorization.

I am clear with 84 = (2)(2)(3)(7) i.e. 7 is the greatest prime factor of 84. So is it correct here that the maximum value of x has to be 84 only, the maximum value of y has to be 84 only and maximum value of z can be 84 only?

So for example if x is 4, y is 84 and z is 84 => LCM of xyz = (2.2)(2.2.3.7)(2.2.3.7) = > Greatest prime factor is 7. Is this approach correct?

Thanks a lot for your help.

Best Regards,
Sri

[Brent@GMATPrepNow]

Hello Brent,

Thanks for your explanation. I just had a question in statement 2:

If 84 is the LOWEST common multiple (LCM), none of the numbers (x, y or z) can have a number bigger than 7 in their prime factorization.

I am clear with 84 = (2)(2)(3)(7) i.e. 7 is the greatest prime factor of 84. So is it correct here that the maximum value of x has to be 84 only, the maximum value of y has to be 84 only and maximum value of z can be 84 only?

So for example if x is 4, y is 84 and z is 84 => LCM of xyz = (2.2)(2.2.3.7)(2.2.3.7) = > Greatest prime factor is 7. Is this approach correct?

Thanks a lot for your help.

Best Regards,
Sri

Yes, that's a great approach.

So is it correct here that the maximum value of x has to be 84 only, the maximum value of y has to be 84 only and maximum value of z can be 84 only?

Yes.
In general, if k is a multiple of n, then n < k
If 84 is the lowest common multiple of x, y and z, then we know that 84 is a multiple of x, 84 is a multiple of y, and 84 is a multiple of x.
So, applying the above rule, we can conclude that x < 84, y < 84, and z < 84

Cheers,
Brent

[Scott@TargetTestPrep]

If x, y, and z are positive integers, what is the greatest prime factor of the product xyz?

(1) The greatest common factor of x, y, and z is 7.

(2) The lowest common multiple of x, y, and z is 84.

Solution:

If the largest prime factors of x, y and z are a, b, and c, respectively, then the greatest prime factor of the product xyz is the largest of the values of a, b and c.

Statement One Only:

The greatest common factor of x, y, and z is 7.

We don’t have enough information to determine the greatest prime factor of xyz. For example, if x = 2 * 7, y = 3 * 7 and z = 5 * 7, then the greatest prime factor of xyz is 7. However, if x = 2 * 7, y = 3 * 7 and z = 7 * 11, then the greatest prime factor of xyz is 11.

Statement Two Only:

The lowest common multiple of x, y, and z is 84.

Since 84 = 2^2 * 3 * 7, it must be true that 7, the greatest prime factor of 84, must be the greatest prime factor of x, y, or z. Therefore, from our stem analysis, 7 is also the greatest prime factor of the product xyz.

Answer: B