# Guest Post: An Interesting Property of Prime Numbers

*Although I have already discussed modulo division, I believe that this proof is beyond the reach of average high school students. To explain further, I made additional notes on Patrick’s proof **. I hope these explanations would be able to help students who want to delve on the proof. *

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I’ve got a prime number trick for you today.

- Choose any prime number .
- Square it.
- Add 5.
- Divide by 8.

Having no idea which prime number you chose, I can tell you this:

The remainder of your result is 6.

Explanation of the Prime Number Trick

We are trying to show that .

This is equivalent to showing that , or that is divisible by . (i)

Because and is prime, then either or . (ii)

Consequently, it must be the case that (a) and or (b) and . (iii)

That is, both numbers will be even, and at least one of them will be a multiple of . For either (a) or (b), the product will be a multiple of . Q.E.D. (iv)

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Notes:

(i) In the congruence, , 6 was just subtracted from both sides of the congruence to obtain . As you can see, congruence works just like equations.

(ii) This explains that any prime number greater than 3 gives remainders of either 1 or 3 when divided by 4. All prime numbers greater than 3 are odd integers, so it cannot give an even remainder when divided by an even number . Can you see why?

(iii) In (a) was added to both sides of and* * and was subtracted from both sides of . In (b), 2 was subtracted from both sides of and 1 was added to both sides of .

(iv) QED stand for *quod erat demonstrandum* which translates to “which was to be demonstrated.” This is usually placed at the end of the proof to signify that the statement above had been proven.

*Author*

*Patrick Vennebush wrote Math Jokes 4 Mathy Folks, a collection of 400+ math jokes. He now shares jokes and blogs about math stuff in his blog Math Jokes for Mathy Folks. *.