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# INSANE ALGEBRA

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How many ordered pairs of positive integers  $(m,n)$ satisfy $\gcd(m,n) = 2$ and $$\mathop{\text{lcm}}[m,n] = 108$$?

Jul 28, 2020

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LCM of 108 and 2 = 108
GCD of 108 and 2 = 2
LCM of 54 and 4 = 108
GCD of 54 and 4 = 2
LCM of 4 and 54 = 108
GCD of 4 and 54 = 2
LCM of 2 and 108 = 108
GCD of 2 and 108 = 2

Note:This question has been posted many times here, but the LCM was 108 and the GCD was 3 NOT 2. if you made a mistake and the GCD should be 3, then:

LCM of 108 and 3 = 108
GCD of 108 and 3 = 3
LCM of 27 and 12 = 108
GCD of 27 and 12 = 3
LCM of 12 and 27 = 108
GCD of 12 and 27 = 3
LCM of 3 and 108 = 108
GCD of 3 and 108 = 3

Jul 28, 2020