The numbers $525$ and $3000$ are both divisible by $3, 5, 15, 25,$ and $75$. What is $HCF(525, 3000)$? Justify your answer.

If $n$ is an odd integer,then show that $n^{2}-1$ is divisible by $8$.

$A$ bookseller has $56$ English books and $72$ Gujarati books. He wants to stack them in such a way that all the books are stored topic-wise and the height of each stack is the same and also the stacks occupy the least surface area of the bottom. What is the maximum number of books that can be placed in each stack for this purpose?

The value of $\text{g.c.d.}(18, 24) \times \text{l.c.m.}(18, 24) = \dots$

Prove that the square of every integer is of the form $3m$ or $3m+1$,where $m \in \mathbb{Z}$.

Prove that the following number is irrational: $\frac{1}{\sqrt{3}}$