If two positive integers $a$ and $b$ are written as $a = x^3 y^2$ and $b = x y^3$,where $x$ and $y$ are prime numbers,then $HCF(a, b)$ is:

The conjugate surd of $3+\sqrt{2}$ is $\ldots \ldots \ldots \ldots .$

Prove that the product of any two consecutive positive integers is divisible by $2$.

Find the $\text{l.c.m.}$ and $\text{g.c.d.}$ of the following by using the fundamental theorem of arithmetic: $84$,$90$,and $120$.

Three sets of English,Hindi,and Mathematics books have to be stacked in such a way that all the books are stored topic-wise and the height of each stack is the same. The number of English books is $96$,the number of Hindi books is $240$,and the number of Mathematics books is $336$. Assuming that the books are of the same thickness,determine the number of stacks of English,Hindi,and Mathematics books.

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