Question

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 $1.$ )

$1,$ $2$

$1,$ $3$

$1,$ $4$

$2,$ $3$

$2,$ $4$

Answer: E

Concepts:greatest common divisor least common multiple prime factorization

Difficulty rating: 1790

Small Hint:

For one prime, order its exponents in $a,b,c$ as $u\le v\le w$

Big Hint:

Compare the exponent $u+w$ in $MD$ with the exponent $u+v+w$ in $abc$

Solution:

For any prime, let its exponents in $a,b,c$ be $u\le v\le w.$ Its exponent in $MD$ is $u+w,$ while its exponent in $abc$ is $u+v+w.$ Thus $MD\le abc,$ proving statement $(2).$ Equality holds exactly when $v=0$ for every prime, meaning no prime divides two of $a,b,c.$ That is precisely pairwise relative primality, proving statement $(4).$

Therefore, the correct answer is E.

1959 AMC 12 Problem 42

Answer: E

Difficulty rating: 1790

Small Hint:

Big Hint:

Solution:

Problem 42 in Other Years