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I framed a new question just now. What is the Remainder when the number $20^{20}$ is divided by $2020$

My try:

$$\frac{20^{20}}{2020}=\frac{20^{19}}{101}$$

Now Consider: $$20^{18}=(400)^9=(404-4)^9=101k-2^{18}$$

Now i was trying to find Remainder without calculator or by manual division.

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    $\begingroup$ It's spelt, "remainder". $\endgroup$
    – Shaun
    Dec 31, 2019 at 18:50
  • $\begingroup$ oops will correct it $\endgroup$ Dec 31, 2019 at 18:52
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    $\begingroup$ Just go by iterated squaring. $20^2=400\equiv -4\pmod {101}$. then $20^4\equiv 16 \pmod {101}$ and so on. $\endgroup$
    – lulu
    Dec 31, 2019 at 18:55
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    $\begingroup$ $20^{20}=10^{20}2^{20}\equiv 100^{10}1024^2\equiv(-1)^{10}{14}^2\equiv196\equiv95\bmod101$ and $20^{20}\equiv0\bmod20$ so $20^{20}\equiv600\bmod2020$ by the Chinese remainder theorem $\endgroup$ Dec 31, 2019 at 21:56

2 Answers 2

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$20^{19}=100^9\cdot4^9\cdot20=100^9\cdot4^{10}\cdot5=100^9\cdot1024^2\cdot5 \equiv - 14^2\cdot5=-980 \equiv 30 \; (\mod 101)$

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    $\begingroup$ That’s quite clever! +1. $\endgroup$
    – ViHdzP
    Dec 31, 2019 at 19:22
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Let’s start with a simpler problem:

What is the remainder of $20^{19}$ when divided by $101$?

We can solve this by Exponentiation by Squares, at each step, just squaring the previous result. This is easy enough to do by hand.

$$20^1\equiv20\pmod{101},$$ $$20^2\equiv97\pmod{101},$$ $$20^4\equiv16\pmod{101},$$ $$20^8\equiv54\pmod{101},$$ $$20^{16}\equiv88\pmod{101}.$$

Since $19=16+2+1$, our desired remainder will be

$$20^{19}=20^{16}\times20^2\times20^1\equiv30\pmod{101}.$$

Finally, using that $a\equiv b\pmod{c}$ iff $ak\equiv bk\pmod{ck}$, for any non-zero $k$, we can deduce

$$20^{20}\equiv\boxed{600}\pmod{2020}.$$

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