2024 AMC 10B Problems/Problem 7

Problem

What is the remainder when is divided by ?

Solution 1

We can factor the expression as

Note that , this expression is actually divisible by 19. The answer is .

Solution 2

If you failed to realize that the expression can be factored, you might also apply modular arithmetic to solve the problem.

Since , the powers of repeat every three terms:

The fact that , , and implies that .

~Bloggish

Solution 3

We start the same as solution 2, and find that:

We know that for , , and , because there are three terms, we can just add them up. , which is mod .

Solution 4 (Given more advanced knowledge)

By Fermat's Little Theorem (FLT), we know that Then its order must divide . Trying simple values we try and succeed:

So the expression is equivalent to , which gives when divided by 19.

~xHypotenuse

Solution 5 (Basic Mod)

7^2024 * (1+7+49) == 0 (mod 19) 57 == 0 mod 19

so it is just .

~Aarav22

🎥✨ Video Solution by Scholars Foundation ➡️ (Easy-to-Understand 💡✔️)

https://youtu.be/T_QESWAKUUk?si=5euBbKNMaYBROuTV&t=100

Video Solution 1 by Pi Academy (Fast and Easy ⚡🚀)

https://youtu.be/QLziG_2e7CY?feature=shared

~ Pi Academy

Video Solution 2 by SpreadTheMathLove

https://www.youtube.com/watch?v=24EZaeAThuE

See also

These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions.