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When the greatest common divisor and least common multiple of two integers are multiplied, their product is 200. How many different values could be the greatest common divisor of the two integers?

Jan 23, 2019

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When the greatest common divisor and least common multiple of two integers are multiplied, their product is 200.
How many different values could be the greatest common divisor of the two integers?

I assume, we have positive integers.

Divisors 200: 1 | 2 | 4 | 5 | 8 | 10 | 20 | 25 | 40 | 50 | 100 | 200 (12 divisors)

$$\begin{array}{|r|r|r|r|r|c|} \hline & a & b & \gcd(a,b) & \operatorname{lcm}(a,b) & \gcd(a,b)\times \operatorname{lcm}(a,b)=a\times b \\ & & & =\gcd(b,a) & =\operatorname{lcm}(b,a) & \\ \hline 1. & 1 & 200 & 1 & 200 & 200 \\ \hline 2. & 2 & 100 & 2 & 100 & 200 \\ \hline 3. & 4 & 50 & 2 & 100 & 200 \\ \hline 4. & 5 & 40 & 5 & 40 & 200 \\ \hline 5. & 8 & 25 & 1 & 200 & 200 \\ \hline 6. & 10 & 20 & 10 & 20 & 200 \\ \hline \end{array}\\ \gcd(a,b)=\{1,2,5,10\}$$

The greatest common divisor could be 4 different values.

Jan 23, 2019
edited by heureka  Jan 23, 2019