Prove that $13$ divides $146^6+1$.
My intuition is to use congruence and find a way to form that into an equation in which we can factor out $13$. Using congruence we get $13 \equiv 146^6+1$ mod($13$) I am not sure where to go from here however.
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Bill Dubuque
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shrizzy
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What does $>\equiv$ mean? – Greg Martin Jan 30 '23 at 05:51
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Compue it $\bmod 13$ using the congruence rules in the linked dupe. – Bill Dubuque Jan 30 '23 at 07:24
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This cannot be proven, because it is actually the case that $13$ does not divide $146^6+1$. Perhaps it is a typo and you meant $146^6-1$?
Using basic modular arithmetic rules: $$146^6\equiv3^6\equiv(3^3)^2\equiv27^2\equiv1^2\equiv1\pmod{13}$$ So, $146^6+1$ is actually equivalent to $2\pmod{13}$, not $0\pmod{13}$ as we would expect if $13$ divides $146^6+1$. You can even verify this answer by typing in "146^6 + 1 mod 13" in Wolfram Alpha.
Now if the problem were asking about $146^6-1$ instead, then it follows from $146^6\equiv1\pmod{13}$ that $146^6-1\equiv0\pmod{13}$.
Aiden Chow
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Please strive not to post more (dupe) answers to dupes of FAQs, cf. recent site policy announcement here. – Bill Dubuque Jan 30 '23 at 07:25
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