Calculate the remainder of $11^{2020} / 7$

Question

I tried to get a pattern by doing

$11^1 / 7 = 1 , r=4$

$11^2/7 = 17, r=2$

$11^3 / 7 = 190, r=1$

but the numbers keep getting larger and larger and I think this is not the way to go about this problem. Can someone please explain the correct way on how to deal with these problems?

Hint: If $11^{3}\equiv 1\pmod{7}$, then $11^{m}\equiv 1\pmod{7}$ for any $m$ that is a multiple of $3$. — Minus One-Twelfth, May 23 '20 at 09:51

score 1 · Accepted Answer · answered May 23 '20 at 09:51

1

$$11^3\equiv 1 \pmod{7}$$

$$11^{2020}\equiv 11^{3\cdot 673}\cdot 11 \equiv (1)^{673}\cdot {11}\equiv 4 \pmod 7$$

answered May 23 '20 at 09:51

h-squared

1,333
6
8

Calculate the remainder of $11^{2020} / 7$

1 Answers1