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shasadou
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Concentration: General Management, Operations
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30 Jan 2016, 12:08
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B
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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
(1) The greatest common factor of x, y, and z is 7.
(2) The lowest common multiple of x, y, and z is 84
30 Jan 2016, 12:13
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Quote:
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
General Discussion
16 Mar 2016, 06:26
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shasadou
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)
