Question 42
Given 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?
$(1)$ 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 $1.$)
$(1)$ 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 $1.$)