Problem Solving

n is a positive integer, what is the remainder when 3^(8n+3) + 2 is divided by 5?

a 0
b 1
c 2
d 3
e 4

The cyclicity of 3's exponents is 4. ie, the units digits are always in the pattern of 3, 9, 7, 1, 3, 9 and so on and so forth.

So multiplying the exponent by 8 and adding 3 to it will always result in the same units digit number.

Eg. if n is 1, then 3^11 will result in 7 as the units digit. Likewise if n is 2, 3, 4, etc.

And 7 + 2 = 9. So the remainder will always be 4.

Answer is E