Question

when the number 3^100 is divided by 5, what is the value of the remainder?

Answer 1

3^1 divided by 5 -> reminder is 3
3^2 divided by 5 -> reminder is 4
3^3 divided by 5 -> reminder is 2
3^4 divided by 5 -> reminder is 1

3^5 divided by 5 -> reminder is 3
3^6 divided by 5 -> reminder is 4
3^7 divided by 5 -> reminder is 2
....
see the regularity? Every 4 numbers, it repeats. it's a set.

SO, 100/4 is 25, an , which means the set repeated 25 times, and ended with the last one on the set. So the reminder is 1

Answer 2

oh wow thanks!

Answer 3

but im still a little confused about why 100/4

Answer 4

If you're familiar with Fermat's Little Theorem:

Since \(100\equiv0\pmod4\) you know that \[3^{100}\equiv(3^4)^{25}\equiv1^{25}\equiv1\pmod5\]So the remainder is 1.

Answer 5

thank you so much!! :D