query: Use group theory to calculate $9^{74} \pmod {13}$.
positive_passages.docid: 4972
positive_passages.text: Mod of numbers with large exponents [modular order reduction]
negative_passages.docid: 3317
negative_passages.text: Subgroup generated by $2$ and $7$ in $(\mathbb Z,+)$
negative_passages.docid: 17609
negative_passages.text: Finding the remainder of $11^{2013}$ divided by $61$
negative_passages.docid: 15897
negative_passages.text: Solve a congruence $1978^{20}\equiv x\pmod{125}$
negative_passages.docid: 48875
negative_passages.text: $n^3 \equiv n^5 \pmod{12} $?
negative_passages.docid: 95653
negative_passages.text: Euler's theorem: [3]^2014^2014 mod 98
negative_passages.docid: 72079
negative_passages.text: What are $2222^{5555}+5555^{2222} \pmod 7$ and $9^{2n+1}+8^{n+2} \pmod{73}$?
negative_passages.docid: 141371
negative_passages.text: Prove that $\alpha^n+\beta^n+\gamma^n \equiv 4^n+5^n+(-6)^n \pmod{17}$
negative_passages.docid: 66161
negative_passages.text: How to solve the equation $n^2 \equiv 0 \pmod{584}$?
negative_passages.docid: 68201
negative_passages.text: Cyclic group of order $n$
negative_passages.docid: 121122
negative_passages.text: Calculate $2^{5104} \bmod 10$ using mental arithmetic
negative_passages.docid: 45947
negative_passages.text: Find generator of multiplicative group of $\mathbb{F}_{27}$
negative_passages.docid: 22783
negative_passages.text: Is the set $Z_n = \{ 0, 1, . . . , n-1 \}$ for $n ≥ 1$ is a group under addition modulo n?
negative_passages.docid: 22084
negative_passages.text: $\mathbb{Z}[i] / (2+3i)$ has 13 elements
negative_passages.docid: 139984
negative_passages.text: Please help me, Group Theory. Prove $b^{33}=e$.
negative_passages.docid: 73020
negative_passages.text: Every integer is congruent to the sum of its digits mod 9
negative_passages.docid: 150007
negative_passages.text: Prove that $(\mathbb{Z}_n , +)$, the integers $\pmod{n}$ under addition, is a group.