What is the remainder obtained when $14^{16}$ is divided with $22$?
Is there a general method for this, without using number theory? I wish to solve this question using binomial theorem only - maybe expressing the numerator as a summation in which most terms are divisible by $22$, except the remainder?
How should I proceed?
You can use binomial expansions and see that $$14^{16} = (22 - 8)^{16}$$ implies that the remainder is just the remainder when $(-8)^{16}( = 8^{16})$ is divided by $22.$ Proceeding similarly,
$8^{16} = 64^8 = (66 - 2)^8 \implies 2^8 = 256 \text{ divided by } 22 \implies \text{remainder = 14}$
Since $$ 14^2 \equiv -2 $$ so $$14^{16} \equiv (-2)^8 \equiv 16^2\equiv (-6)^2 \equiv 14$$
or
$$ 14^2 = 22k -2 $$ so $$14^{16} = (22k-2)^8 = 22l+2^8=22l+22\cdot 11+14$$
A method that uses FLT.
Finding the remainder on dividing $14^{16}$ by $22$ is equivalent to twice that on dividing $7\cdot14^{15}$ by $11$.
By FLT, $14^{10}\equiv1\pmod{11}$ so $$7\cdot14^{15}\equiv7\cdot14^5\equiv98\cdot196\cdot196\equiv-1\cdot(-2)\cdot(-2)\equiv-4\equiv7\pmod{11}$$ hence the required remainder is $7\times 2=14$.
Using the Euclidean Algorithm
Note that $$ \begin{align} 14^{16}&\equiv0&\pmod2\tag1 \end{align} $$ Reducing mod $11$ and using Fermat's Little Theorem, we get $$ \begin{align} 14^{16} &\equiv3^6&\pmod{11}\\ &\equiv3&\pmod{11}\tag2 \end{align} $$ Solving $11a+2b=1$ using the Euclidean Algorithm as implemented in this answer, $$ \begin{array}{r} &&5&2\\\hline 1&0&1&-2\\ 0&1&-5&11\\ 11&2&1&0\\ \end{array}\tag3 $$ we have $$ 11(1)+2(-5)=1\tag4 $$ from which we get $$ \begin{align} -10&\equiv0&\pmod2\\ -10&\equiv1&\pmod{11} \end{align}\tag5 $$ Multiplying $(5)$ by $3$ gives $$ \begin{align} -30&\equiv0&\pmod2\\ -30&\equiv3&\pmod{11} \end{align}\tag6 $$
Using the Chinese Remainder Theorem
The Chinese Remainder Theorem says that the solution to $(6)$ is unique mod ${22}$. Thus, we get the smallest positive solution to be $$ \begin{align} 14&\equiv0&\pmod2\\ 14&\equiv3&\pmod{11} \end{align}\tag7 $$ Thus, $(1)$, $(2)$, and $(7)$ yields $$ \begin{align} 14^{16}&\equiv14&\pmod{22}\tag8 \end{align} $$
