Problem: Given three positive integers , and . Their greatest common divisor is ; their least common multiple is . Then, which two of the following statements are true?
(1) The product MD cannot be less than abc.
(2) The product MD cannot be greater than abc.
(3) MD equals abc if and only if a,b,c are each prime.
(4) MD equals abc if and only if a,b,c are relatively prime in pairs,(This means: no two have a common factor greater than 1)
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Represent in terms of their prime factors. Then is the product of all the common prime factors, each factor taken as often as it appears the least number of times in a or or is the product of all the non-common prime factors, each factor taken as often as it appears the greatest number of times in a or or .
Therefore, may be less than , but it cannot exceed .
Obviously, equals abe when there are no common factors.