Prove these numbers are composite: $500^{27}-123^{12}$, $12^{75}+21^{75}$, $58^{32}-49^{32}$

Question

I have to prove the next 3 numbers are composite numbers

$500^{27}-123^{12}$

$12^{75}+21^{75}$

$58^{32}-49^{32}$

I have no idea where to begin, if anyone can give me a hint I will be grateful.

Thank You Very Much!

Well, the usual way to start would be to look for small prime factors. Or to search for "plynomial" factorings, as in $a^2-b^2=(a+b)(a-b)$ — lulu, Mar 25 '21 at 10:51
Please see how to ask a good question. It doesn't have to be what you have tried: for example, where does this question come from? — Toby Mak, Mar 25 '21 at 10:53
This is a test of your algebraic skills in disguise. Your question is equivalent to showing that $a^{27} - b^{12}$, $m^{75} + n^{75}$ and $p^{32} - q^{32}$ factor. — Toby Mak, Mar 25 '21 at 10:58

score 1 · Accepted Answer · answered Mar 25 '21 at 10:56

1

Hint:

Can you write this in the form $a^3-b^3$? Now recall the formula for the difference of two cubes.
$\gcd(12, 21)= 3$, so you can find a common factor.
Use the difference of two squares.

answered Mar 25 '21 at 10:56

Toby Mak

Thank you very much, now I see its easy! – JollyQ1 Mar 25 '21 at 12:09
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Please strive not to add more dupe answers to dupes of FAQs. – Bill Dubuque Mar 26 '21 at 09:19
Will do, thank you. – Toby Mak Mar 26 '21 at 10:26

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