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
Answers
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.