1959 AHSME Problems/Problem 42

Problem

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? $\text{(1)}\ \text{the product MD cannot be less than abc} \qquad \\ \text{(2)}\ \text{the product MD cannot be greater than abc}\qquad \\ \text{(3)}\ \text{MD equals abc if and only if a,b,c are each prime}\qquad \\ \text{(4)}\ \text{MD equals abc if and only if a,b,c are each relatively prime in pairs} \text{ (This means: no two have a common factor greater than 1.)}$ $\textbf{(A)}\ 1,2 \qquad\textbf{(B)}\ 1,3\qquad\textbf{(C)}\ 1,4\qquad\textbf{(D)}\ 2,3\qquad\textbf{(E)}\ 2,4$

Solution

Because $1\times2\times4>1\times4$ , 1 is false. Because $1\times1\times1=1\times1$ , 3 is false. It follows that the answer is $\boxed{\textbf{E}}$ .

1959 AHSC (Problems • Answer Key • Resources)
Preceded by Problem 41	Followed by Problem 43
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1959 AHSME Problems/Problem 42

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