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I am writng some material about congruences. I'm wondering if it is correct to write it like this:

\begin{align*} 2^{30}&\equiv 2^{6}(\operatorname{mod}13)= 2^{4}\cdot 2^{2}(\operatorname{mod}13) \equiv 16\cdot 4\,(\operatorname{mod}13)\\ &\equiv 3\cdot 4(\operatorname{mod}13)= 12 \,(\operatorname{mod}13)\equiv (-1)\,(\operatorname{mod}13) \end{align*}

Primarily, I'm thinking if it is ok to write signs = and $\equiv $ in the same line and am I writing $(\operatorname{mod}13) $ too many times? Should I write it only once at the end of calculus? I have seen so many ways of writing it and I am not sure what is the best. Thank you all.

Bill Dubuque
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joanna
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1 Answers1

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Indeed, I also would prefer that $=$ and $\equiv$ are not mixed here. Also, it is enough to write modulo $13$ at the end: $$ 2^{30} \equiv 2^{6}\equiv 2^{4}\cdot 2^{2}\equiv 16\cdot 4\equiv 3\cdot 4 \equiv -1 \pmod{13} $$ Here we have used $2^{12}\equiv 1\pmod{13}$, by Little Fermat.

Dietrich Burde
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  • The bmod above is incorrect and pedagogically misleading (it promotes the common confusion between mod operations vs. relations). Also "should not" is too strong, since sometimes using an equality can convey useful info about the implicit reasoning employed – Bill Dubuque Nov 26 '24 at 10:07
  • @BillDubuque I see. I replaced it by pmod. – Dietrich Burde Nov 26 '24 at 10:24
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    Please strive not to post more (dupe) answers to dupes of FAQs. This is enforced site policy, see here. – Bill Dubuque Nov 26 '24 at 10:29