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I am studying basic number theory and have a habit of writing down interesting facts whenever I conclude something from the text or a problem itself. I was wondering whether I can write it down too:

All odd prime numbers other than 5, either themselves are one less or more than a multiple of 10, or their square is 1 less than a multiple of 10.

What do you say?

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1 Answer 1

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Yes, you don't even need the numbers to be prime. This can be proven via basic modular arithmetic. All $1,3,7,9 \mod 10$ follow the above rules.

Edit:- So you don't need modular arithmetic, but it does make your life easier. First, note that if any odd number ends in $1$ or $9$ then the first property is satisfied. Second, if it ends in $3$ or $7$, write it as $10n+3$ or $10n+7$ and square it to conclude that second property is satisfied

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  • $\begingroup$ Thanks for the confirmation Anvit. I have yet to study Modular Arithmetic, but I'll keep this in mind. $\endgroup$ Jun 1, 2020 at 11:58
  • $\begingroup$ @AdilMayank I've added answer w/o modular arithmetic $\endgroup$
    – Anvit
    Jun 4, 2020 at 7:42
  • $\begingroup$ Thanks again, this explanation was more relatable. $\endgroup$ Jun 18, 2020 at 16:07

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