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When the greatest common divisor and least common multiple of two integers are multiplied, the product is 180. How many different values could be the greatest common divisor of the two integers?

Jun 23, 2018

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When the greatest common divisor and least common multiple of two integers are multiplied, the product is 180.

How many different values could be the greatest common divisor of the two integers?

$$\begin{array}{|rcr|r|r|} \hline 180 &=& 180 \times 1 & lcm(180,1) = 180 & gcd(180,1) = 1 \\ 180 &=& 90 \times 2 & lcm( 90,2) = 90 & gcd( 90,2) = 2 \\ 180 &=& 60 \times 3 & lcm( 60,3) = 60 & gcd( 60,3) = 3 \\ 180 &=& 45 \times 4 & lcm( 45,4) = 45 & gcd( 45,4) = 1 \\ 180 &=& 36 \times 5 & lcm( 36,5) = 36 & gcd( 36,5) = 1 \\ \hline \end{array}$$

The greatest common divisor of the two integers could be  3 different values (1,2, and 3)

Jun 25, 2018