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

2024/056) How do I find the minimum positive integer $n>114$, such that $2001 \equiv 114(\mod n)$

It is 114 mod n so 2001–114 should be divisible by n and $n > 114$

Factors of 1887 are 1 3 17 37 51 111 629 1887. any number $<= 111$ shall give a smaller remainder and 2 numbers 629 and 1187 shall give a remainder 114 so the number is lower of the 2 that is 629.