Find the remainder when 3^997

Question

is divided by 5.

n=1 3n/5=3/5
n=2 3^2=9/25
n=3 3^3=27/5
n=4 3^4=81/5
n=5 3^5=243/5

r=3,4,2,1,3,4,2,1
n=997

Answer 1

Note that the last digit of 3^n repeats:

3,9,7,1,...
3^997 = 3^996*3, so it ends in 3
so, ...

Answer 2

you were on the right track.

look at these two columns

n r , where r is the remainder after dividing by 5
1 3
2 4
3 2
4 1
5 3
6 4
7 2
8 1
9 3
10 4
11 2
12 1

So the remainders cycle as 3 4 2 1,

so if n is evenly divisible by 4 , the remainder of 3^n divided by 5 will be 1
if n divided by 4 leaves a remainder of 1, such as n=5 or n=9 , the remainder of 3^n divided by 5 will be 3

if n divided by 4 leaves a remainder of 2, such as n=6 or n=10 , the remainder of 3^n divided by 5 will be 4

if n divided by 4 leaves a remainder of 3, such as n=7 or n=11 , the remainder of 3^n divided by 5 will be 2

So 997 ÷ 4 leaves a remainder of 1
so 3^997 leaves a remainder of 3 when divided by 5