query: Why is true that $(a^b\bmod k)^c \bmod k = \left(a^b\right)^c \bmod k$?
positive_passages.docid: 10105
positive_passages.text: Proving $\, a^b = (a\bmod n)^b\, \pmod{\!n}\ $ [Congruence Power Rule]
negative_passages.docid: 125868
negative_passages.text: Prove $a \equiv b \pmod{c} \implies a^n \equiv b^n \pmod{c}$.
negative_passages.docid: 48211
negative_passages.text: When is $(a+b)^n \equiv a^n+b^n$?
negative_passages.docid: 18714
negative_passages.text: $\gcd\left(a+b,\frac{a^p+b^p}{a+b}\right)=1$, or $p$
negative_passages.docid: 157087
negative_passages.text: Can anyone explain why $a^{b^c} = a^{(b^c)} \neq (a^b)^c = a^{(bc)}$
negative_passages.docid: 75199
negative_passages.text: $a \equiv b \pmod n$ and $c\equiv d \pmod n$ implies $ac \equiv bd \pmod n$
negative_passages.docid: 100303
negative_passages.text: A necessary and sufficient condition for $A\cap(B+C)=(A\cap B)+C$ to hold for submodules
negative_passages.docid: 15626
negative_passages.text: Relations of a~b iff b = ak^2
negative_passages.docid: 111505
negative_passages.text: Proof $\binom{p-1}{k}\equiv (-1)^k \mod{p}$
negative_passages.docid: 136185
negative_passages.text: How $a^{\log_b x} = x^{\log_b a}$?
negative_passages.docid: 67660
negative_passages.text: Prove $\,b\equiv c\pmod{\!m}\Rightarrow \gcd(b,m)=\gcd(c,m)$ i.e. $\gcd(c+km,m) = \gcd(c,m)$
negative_passages.docid: 147665
negative_passages.text: $a_1=3;\ a_{n+1}=3^{a_n}$; find $a_{2004}\bmod100$
negative_passages.docid: 34714
negative_passages.text: What is the relationship between $\ker(A), \ker(A^2), \ldots, \ker(A^n)$?
negative_passages.docid: 26301
negative_passages.text: Prove that $x^3 \equiv x \bmod 6$ for all integers $x$
negative_passages.docid: 120911
negative_passages.text: Clarification on proof: Order of left cosets equal
negative_passages.docid: 97184
negative_passages.text: Show: $ f(a) = a,\ f(b) = b \implies \int_a^b \left[ f(x) + f^{-1}(x) \right] \, \mathrm{d}x = b^2 - a^2 $
negative_passages.docid: 111555
negative_passages.text: Find $a,b \in \mathbb{N}$ with $\mathrm{lcm}(a,b)=12\gcd(a,b)\,$ and $\,a\bmod b = 5$