Proof concerning prime factorizations

Question

I'm trying to show a relationship of a number and its prime factors. The problem follows as:

Show if n >= 2 and n is not an element of the primes, there exists a prime number p such that p | n and p <= root n.

I'm coming near some kind of point to start from but can't make that initial step into it, any tips? Thanks.

An example of the problem

Answer 1

Since $n > 1$ is not prime, we may write $n = ab$ where $a,b > 1$ are positive integers. If both $a,b$ were greater than $\sqrt{a}$, then $ab > a$, a contradiction. Thus at least one is less than or equal to $\sqrt{n}$, say $a$, and by the Fundamental Theorem of Arithmetic $a$ is a product of primes, one of which is less than or equal to $\sqrt{n}$.

Answer 2

Taking as a counterexample: $n=10 \Rightarrow n = 2\cdot 5$

Then we have that $2 \not | \sqrt{10}$ and $5 \not | \sqrt{10}$ and hence the proposition is false.