Question

Given are three positive integers $a,b,$ and $c$. Their greatest common divisor is $D$; their least common multiple is $M$. Then, which two of the following statements are true?
$\left(1\right)$ The product $MD$ cannot be less than $abc$
$\left(2\right)$ The product $MD$ cannot be greater than $abc$
$\left(3\right)$ $MD$ equals $abc$ if and only if $a,b,c$ are each prime
$\left(4\right)$ $MD$ equals $abc$ if and only if $a,b,c$ are relatively prime in pairs
(This means : no two have a common factor greater than $1$.)

• A. $1,2$
• B. $1,3$
• C. $1,4$
• D. $2,3$
• E. $2,4$

Answer 1

Answer: (E)

$2,4$

Represent $a,b,c$ in terms of their prime factors. Then $D$ 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 $b$ or $c$. $M$ is the product of all the non-common prime factors, each factor taken as often as it appears the greatest number of times in $a$ of $b$ or $c$.
Therefore, $MD$ may be less than $abc$, but it cannot exceed $abc$.
Obviously, $MD$ equals $abc$ when there are no common factors.