Remainder with 2018?

Number Theory Level 3

What is the remainder when $2018^{2018}$ is divided by 20?

5 2 1 3 4

1 solution

Mark Hennings
Aug 2, 2018

Certainly $2018^{2018} \equiv 0 \pmod{4}$ . Also $2018^{2018} \equiv 3^{2018} \equiv (-1)^{1009} \equiv -1 \equiv 4 \pmod{5}$ . Solving the suitable Chinese Remainder problem, we deduce that $2018^{2018} \equiv \boxed{4} \pmod{20}$ .

How are you "certain" that $2018^{2018} \equiv 0 \pmod {4}$ ? Can you give an in-detail solution for your other steps too?

Syed Hamza Khalid - 2 years, 10 months ago

More than two copies of a multiple of $2$ must multiply to a multiple of $4$ . As to the rest, you need to think about clock arithmetic modulo $5$ .

Mark Hennings - 2 years, 10 months ago

Hey Mark Hennings, can you help me for the following:

https://brilliant.org/discussions/thread/some-problems-which-i-am-stuck-at-any-help-i-beg/

Syed Hamza Khalid - 2 years, 10 months ago

Remainder with 2018?

1 solution

0 pending reports