$29 \cdot 31$ Given three positive integers $a, b, c$, their greatest common divisor is $D$, and their least common multiple is $M$. Among the following statements, which are true:
(1) The product $M D$ is not less than $a b c$.
(2) The product $M D$ is not greater than $a b c$.
(3) $M D = a b c$ if and only if $a, b, c$ are prime numbers.
(4) $M D = a b c$ if and only if $a, b, c$ are pairwise coprime.
(A) 1,2 .
(B) 1,3 .
(C) 1,4 .
(D) 2,3 .
(E) 2,4 .
(10th American High School Mathematics Examination, 1959)
[Solution] When $a, b, c$ are expressed as products of prime factors:
$D$ is the product of the factors with the lowest powers;
$M$ is the product of the factors with the highest powers.
Therefore, the power of each factor in $a b c$ is not less than the power of the corresponding factor in $M D$, and any factor in $M D$ must also be in $a b c$, so $M D \leqslant a b c$.
However, when $a, b, c$ are coprime, $M D = a b c$.
Therefore, the correct choice is $(E)$.