2024/041) The GCD of 3 numbers is 30 and their LCM is 900. Two of the numbers are 60 and 150. What is the possible third number?

We have factoring all 4

$60 =  2^2 * 3 *5 $

$150 = 2 * 3 * 5^2$

$30 = 2 * 3 * 5$

$900 = 2^2 * 3^2 * 5^2$

Because the  GCD is 30 so the number has to be multiple of 30.

Now let us consider highest power of 2 coming from the numbers 900 has 2 and 60 has so 3rd number need not have (might have but not necessary)

Consider highest power of 3 coming from the numbers 900 has 3 and 60 and 150 have 2 so 3rd number must have so it has to be multiple of $2 * 3^2 * 5$ that is 90 

Consider highest power of 5 coming from the numbers 900 has 2  and 150  has so 3rd number need not have (might have but not necessary)

So 3rd number has to be multiple of 90 and it has to de divisor of 900 say 90m

90m is a factor of 900 or  is a factor of 10 that is 1 or 2 or 5 or 10

giving 3rd number  one of 90,180,450,900