The least number exactly divisible by $36$ and $24$ is

If $x$ is prime, $y$ is a composite number such that $x+y=240$ and their $\mathrm{LCM}$ is $4199$. Find $x$ and $y$.

The $\mathrm{LCM}$of two numbers is $1024$ and one of them is a prime number. Find their $\mathrm{HCF}$.

The $\mathrm{HCF}$of two numbers is $4$ and their $\mathrm{LCM}$  is $400$. How many pairs of values can the numbers assume?

Find the greatest number that can divide $101$ and $115$ leaving remainders $5$ and $7$ respectively.

Find the least three digit number which when divided by $20, 30, 40$ and $50$ leaves remainder $10$ in each case.

Find the largest number that divides $59$ and $54$ leaving remainders $3$ and $5$ respectively

Can two numbers have $12$ as their $\mathrm{HCF}$ and $512$ as their $\mathrm{LCM}$? Justify your answer