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gcd(a,b,c)lcm(a,b,c)

Source: 1959 AHSME Problem 42

August 14, 2013
number theoryrelatively primegreatest common divisorleast common multipleAMC

Problem Statement

Given three positive integers a,b,a,b, and cc. Their greatest common divisor is DD; their least common multiple is mm. Then, which two of the following statements are true? (1) the product MD cannot be less than abc \text{(1)}\ \text{the product MD cannot be less than abc} \qquad (2) the product MD cannot be greater than abc\text{(2)}\ \text{the product MD cannot be greater than abc}\qquad (3) MD equals abc if and only if a,b,c are each prime\text{(3)}\ \text{MD equals abc if and only if a,b,c are each prime}\qquad (4) MD equals abc if and only if a,b,c are each relatively prime in pairs\text{(4)}\ \text{MD equals abc if and only if a,b,c are each relatively prime in pairs}  (This means: no two have a common factor greater than 1.)\text{ (This means: no two have a common factor greater than 1.)} <spanclass=′latex−bold′>(A)</span> 1,2<spanclass=′latex−bold′>(B)</span> 1,3<spanclass=′latex−bold′>(C)</span> 1,4<spanclass=′latex−bold′>(D)</span> 2,3<spanclass=′latex−bold′>(E)</span> 2,4 <span class='latex-bold'>(A)</span>\ 1,2 \qquad<span class='latex-bold'>(B)</span>\ 1,3\qquad<span class='latex-bold'>(C)</span>\ 1,4\qquad<span class='latex-bold'>(D)</span>\ 2,3\qquad<span class='latex-bold'>(E)</span>\ 2,4
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