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This question already has an answer here:

The problem is following:

Given that $p$ is a prime number, $p > 3$. Prove that $(p^2 - 1)$ is divisible by $24$.

I started writing down the possible remainders of dividing $p$ by $24$ and got the following row: ${5; 7; 11; 13; 17; 19; 23}$

But am i even right at this point? If i am how do i prove that these are the only possible remainders?

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marked as duplicate by Jyrki Lahtonen, Community Oct 14 at 18:58

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You have missed that $1$ can be a remainder too (take $p=73$ or $p=97$, for instance). All other elements of $\{0,1,2,\ldots,23\}$ are muliples of $2$ or of $3$. Since $p$ is prime and $24$ is a multiple of both $2$ and $3$, no such number can be the remainder of the division of $p$ by $24$, since that would make $p$ a multiple of $2$ or of $3$.

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