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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

The problem

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  • $\begingroup$ Taking $n = 10$, doesn't your statement fail? $\endgroup$
    – mrnovice
    Mar 19, 2017 at 18:17
  • $\begingroup$ Not true as $n=6$ fails. $\endgroup$
    – Sentinel135
    Mar 19, 2017 at 18:17
  • $\begingroup$ I may have stated the problem incorrectly, take a look at the picture of it I have posted $\endgroup$
    – Howard P
    Mar 19, 2017 at 20:03

2 Answers 2

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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 $\sqrt{n}$.

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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.

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  • $\begingroup$ Hi, thanks for responding so quickly, I may have stated the problem incorrectly try taking a look at the picture of it I posted $\endgroup$
    – Howard P
    Mar 19, 2017 at 20:04

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