2024/057) The LCM of three numbers is 72 and their GCF is 6. If the first and second number is 12 and 18 respectively, what is the third number?

We have GCD = $6 = 2 * 3$

$1^{st}   number = $12 = 2^2  *3$

$2^{nd}$  number =  $18= 2 * 3^2$

LCM = $72 = 2^3 * 3^2$

So the $3^{rd}$ number has to of the form $2^a3^b$

Taking the lowest powers of 2 and 3 we get the HCF 

So HCF = $2^{min(1,a)}3^{min(1,b)}= 2 * 3$ which gives $a >=1$ and $b >=1$

Taking the highest  powers of 2 and 3 we get the LCM  

 LCM=  $2^{max(2,a)}3^{max(2,b)}= 2^3 * 3^2$ so $a=3$ and $ b <=2$

Giving $a=3,b=1$ that is number = $2^3*3=24$ or $a=3,b=3$ that is number = $2^3*3^2=72$

  so $3^{rd}$ number can be 24 or 72