About question of fundamental theorem of arithmetic

Question

A product of nonzero integers whose absolute values are $<p$ will have the property that all its prime factors are $≤p−1$.

I know that Composite Number has Prime Factor not Greater Than its Square Root. But how this solve the above problem.

Answer 1

It suffices to prove this for nonnegative numbers. Note that $a, b < p \implies a, b \leq p-1$. Try to look at an equal factorization in power of primes (nonnegative exponents), and let $p_k$ be the greatest prime factor of either $a$ or $b$. Then by the Fundamental Theorem of Arithmetic, their products have the same greatest prime factor, which either satisfies $p_k = p-1$ or $p_k \leq \sqrt{p-1} \leq p-1$. Hence all the prime factors $p_i$ are less or equal to $p-1$.