What is the remainder when $7^{113}$ is divided by $50$?

Question

What is the remainder when $7^{113}$ is divided by $50$? is there any method to approach such question

Possible duplicate of How do I compute $a^b,\bmod c$ by hand? — Eric Wofsey, Sep 15 '19 at 22:04
No, there is no method to approach such a question. That's why no teacher has ever asked a class to solve such a question. — Gerry Myerson, Sep 16 '19 at 23:05

score 3 · Answer 1 · edited Sep 15 '19 at 05:50

3

We know that $7^2=49 \equiv -1 \pmod{50}$, so we can write, that:

$$7^{113}=7 \times (7^2)^{56} \equiv 7 \times (-1)^{56} \equiv 7 \pmod{50}$$

edited Sep 15 '19 at 05:50

Henno Brandsma

answered Sep 15 '19 at 05:29

Andronicus

1

\pmod{50} is the correct operator for what you want to achieve; the p in pmod means adding extra parentheses the correct way (this is how I remember it). – Henno Brandsma Sep 15 '19 at 05:52
@HennoBrandsma thank you very much, I'll remember it now! – Andronicus Sep 15 '19 at 05:54

1 Answers1