Express $5005$ as a product of its prime factors.

Prove that $\sqrt{3}$ is irrational.

Find the $LCM$ and $HCF$ of the following pairs of integers and verify that $LCM \times HCF =$ product of the two numbers $26$ and $91$.

Write down the decimal expansion of the rational number $\frac{129}{2^{2} 5^{7} 7^{5}}$.

An army contingent of $616$ members is to march behind an army band of $32$ members in a parade. The two groups are to march in the same number of columns. What is the maximum number of columns in which they can march?

Prove that $7 \sqrt{5}$ is an irrational number.