If x, y, and z are positive integers, what is the greatest prime facto

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
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Prime factors of xyz are prime numbers which are factors of at least one of x, y and z.
By multiplying x, y and z, you will not get a new prime factor because a prime number cannot be obtained by multiplying two positive integers (other than 1 and itself).

So "what is the greatest prime factor of the product xyz" basically asks you for the greatest prime factor out of all prime factors of x, y and z.

(1) The greatest common factor of x, y, and z is 7.
7 is the greatest COMMON factor. So all x, y and z have 7. But one or two of them could have a higher prime factor too such as 11 or 13 etc. We don't know the greatest prime factor out of all prime factors of these integers. Not sufficient.

(2) The lowest common multiple of x, y, and z is 84
84 = 2*2*3*7
These prime factors include all prime factors which are present in each of x, y and z. The greatest among them is 7. So 7 is the greatest prime factor out of all prime factors of these integers. Sufficient.

Answer (B)
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Excellent Question.
Testing our knowledge about GCD-LCM and Prime factorisation.

Given data -> x,y,z are positive integers.
We are asked about the Greatest prime factor of x*y*z

Statement 1->
GCD(x,y,z)=7
Lets use test cases->
Case 1=>
7
7
7
GCD=> 7
And Greatest prime =7
Case 2=>
7
7
7*13
GCD=>7
Greatest prime factor is 7*13

Actually this statement tells us that the greatest prime factor must be greater than or equal to 7.
Hence not sufficient.

Statement 2->
LCM(x,y,z)=84=2^2*3*7
Now if x,y,z had any prime factor other than 2,3,7 => It must have been present in the LCM.
In LCM => We pick the greatest power of each Prime factor.
Hence the greatest prime in the Product must be 7.
Hence sufficient.
Hence B.
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Bunuel: Can you please provide similar questions to practice.

Please find explanation as mentioned in attachment

Answer: option B

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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
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