How do you in general calculate the period of a number $x^m$ modulo $n$ using the Chinese Remainder Theorem if $n$ is squarefree? We see that it is sufficient to find the residues modulo every prime factor of $n = a_1 a_2 a_3 \cdots$ where $a_1,a_2,\ldots$ are the prime factors of $n$ listed in a...

$A$ is a nonsymmetric matrix. $\beta_k, \gamma_k$ are sets of permutations. Each permutation acts on column of $A$. $\beta_k, \gamma_k$ have same permutations but different labeling is used to label columns of $A$. $\beta_k$ used one kind of labeling, $ \gamma_k$ used a different labeling. Con...

$$ \frac{\int_0^{4\pi} e^t(\sin^6at+\cos^4at)\,dt}{\int_0^\pi e^t(\sin^6at+\cos^4at)\,dt}= L, $$ the question asks the value of $a$ and $L$. My friend solved it by differentiating, but i didn't understood a single thing. May be it is a very simple one, but i don't know how to solve it.

I saw this question and wondered how OP of that question was able to do : $$0<\sin x+1<2$$ this $$\frac 0{|x|}<\frac{\sin x+1}{|x|}<\frac 2{|x|}$$ and when $x\to \infty$ he got the limit evaluated as zero. Why i wondered is because it is not working on this inequality $$2\leq x+ \frac 1x\leq 20...