Prove that,if a positive integer is of the form $6m + 5$,then it is also of the form $3n + 2$ for some integer $n$.

$A$ sweet seller has $420$ Kaju burfis and $130$ Badam burfis. He wants to stack them in such a way that each stack carries the same number of items and they take up the least area of the tray. What is the maximum number of burfis that can be placed in each stack for this purpose?

Prove that one and only one out of $n, n+2, n+4$ is divisible by $3$,where $n$ is any positive integer.

Find the square root of $6+\sqrt{35}$ in the form of a binomial surd.

If $a$ and $b$ are two odd positive integers such that $a > b,$ then prove that one of the two numbers $\frac{a+b}{2}$ and $\frac{a-b}{2}$ is odd and other is even.

Find the $\text{l.c.m.}$ and $\text{g.c.d.}$ of the following by using the fundamental theorem of arithmetic: $144$,$180$,and $192$.