Задача 16.

Show that 1) any number that is bigger that $1$ can be expressed as a product of prime numbers; 2) a number $x>1$ can be expressed as \begin{equation*} x=p_1^{a_1}p_2^{a_2}\ldots p_n^{a_n}, \end{equation*} where $p_1< p_2< \ldots< p_n$ are prime numbers and $a_1, a_2,\ldots, a_n$ positive integer numbers. Such expression of a positive integer is called the canonical representation; 3*) (Fundamental theorem of arithmetic) if $x$ is expressed in the following ways \begin{equation*} x=p_1^{a_1}p_2^{a_2}\ldots p_n^{a_n}= q_1^{b_1}q_2^{b_2}\ldots q_m^{b_m}, \end{equation*} then $m=n$, and for any $1\leqslant i\leqslant n$ $p_i=q_i$, $a_i=b_i$; 4) if in the above expression each $a_i$ is even, then $x$ is a square of some number, i.e. there is $y$ such that $x=y^2$.