How to efficiently compute $\,25^{37}\pmod{55}$?

Question

Most efficient way to do this question?

$25^{37}\cong (mod 55)$

Attempt:

$25^2 \cong 20 (mod55)$

Then I don't know the best way to continue... I know how to brute force it but I'm just curious what the best way is?

Answer 1

First, $55 = 5 \cdot 11$ and $(5,11) = 1$, so we will use the Chinese remainder theorem to assemble the result after working modulo $5$ and $11$, separately.

Modulo $5$, $25 \cong 0$, so $25^{37} \cong 0 \pmod{5}$.

Modulo $11$, $25 \cong 3$ and (using Fermat's little theorem) $37 \cong 7 \pmod{10}$, so we compute $$ 25^{37} \cong 3^{37} \cong 3^7 \pmod{11} \text{.} $$ Then \begin{align*} 3^{7} &\cong 3^8 \cdot 3^{-1} \pmod{11} \\ &\cong ((3^2)^2)^2 \cdot 3^{-1} \pmod{11} \\ &\cong (9^2)^2 \cdot 3^{-1} \pmod{11} \\ &\cong 4^2 \cdot 3^{-1} \pmod{11} \\ &\cong 5 \cdot 3^{-1} \pmod{11} \\ &\cong 5 \cdot 4 \pmod{11} \\ &\cong 9 \pmod{11} \text{.} \end{align*}

Then, using the CRT, we want a multiple of $5$ that is congruent to $9$ modulo $11$. Check sequentially: $5$, $10$, $15$, $20$. We find that $20$ works. (There are direct methods, but it's easy to work through the (at most) eleven multiples of $5$.)

Our answer is $25^{37} \cong 20 \pmod{55}$.

Answer 2

You can use the Chinese Remainder Theorem, and Fermat's Little Theorem. Observe that $55=5\times 11$. Clearly $25^{37}\equiv0\pmod5$. Also $$25^{37}=5^{74}\equiv 5^4=25^2\equiv 3^2\equiv 9\pmod{11}$$ using $5^{10}\equiv1\pmod{11}$ (Fermat's Little Theorem). Using the Chinese Remainder Theorem one gets $25^{37}\equiv20\pmod{55}$.

Answer 3

This answer is not as slick as the answers posted before mine. But, this technique works for incredibly large powers even when you do NOT know how the modulus factors...

$37 = 1 + 4 + 32 = 2^0 + 2^2 + 2^5$ so record $25^{2^k} \bmod 55$ for $k=0, 1, \ldots, 5$, but get it by repeated squaring, which is fast: $25^{2^{k+1}}=(25^{2^k})^2$. Get, in order, $$ 25, 20, 15, 5, 25, 20. $$ Now multiply the entries corresponding to $k=0, 2, 5$ to get $$ 25 \cdot 15 \cdot 20 = 20. $$ Hence $25^{37} = 20 \bmod 55$.

Answer 4

$$25^{37}=5^{74}$$

Now as $(5^n,55)=5$ for $n\ge1$

$$5^{74-1}\equiv5^{73\pmod{\phi(55/5)}}\pmod{55/5}$$

$$\implies5^{73}\equiv5^3\equiv4\pmod{11}$$

$$\implies5\cdot5^{73}\equiv5\cdot4\pmod{5\cdot11}$$

Answer 5

Note $\,5^{\large 74} \bmod 55 = \overbrace{5(5^{\large 3}(\color{#c00}{5^{\large 10}})^{\large 7} \bmod 11)}^{\Large 1\ \equiv\ \color{#c00}{5^{\Large 10}} \ {\rm by \ Euler}} = 5(4)\ $

using $\,\ ab\bmod ac = a(b\bmod c) =$ mod Distributive Law to pull a factor of $\,5\,$ out of the $\!\bmod$