The GCF of three numbers is 6 and their LCM is 900. If two of the numbers are 36 and 60, find the other number.

Hello!

Denote the unknown number as `x.` Factor the given numbers into primes:

`36 = 2^2*3^2,`  `60 = 2^2*3*5,`  `6 = 2*3,`  `900 = 2^2*3^2*5^2.`

The `GCF(36,60)=2^2*3=12,`  not `6.` Therefore to have `GCF(36,60,x) = 6,` `x` must have `6` as its factor, but not have `12` as its factor.

Further, the `LCM(36,60)=2^2*3^2*5=180.`

The given `LCM(36,60,x) = 900 = 180*5,` so `x` must have one more factor of `5` than `60,` i.e. it must have a factor of `5^2 = 25.`

This way we know that `x` must have `6*25 = 150` as its factor, so `x = 150*y,` and `y` must be odd. From the other hand, `x` must be a factor of its multiple `900,` so y must be a factor of `900/150=6,` and not so many variants remain.

Actually they are `y=1,` `x=150` and `y=3,` `x=450.`

There are two possible answers: 150 and 450 (not counting -150 and -450).