What is the remainder when 32^32^32 is divided by 9?

Question

A positive integer N leaves the same remainder of 35 when divided by both 2009 and 2010.

What is the remainder when N is divided by 42?

Answer 1

Answer : 4

Solution:
(32^32^32)mod9 = ((-4)^32^32))mod 9

= (4^32^32)mod 9 (given that minus to even power is positive) ---- (i)

Now
(4^1)mod9 = 4
(4^2)mod9 = 7
(4^3)mod9 = 1
(4^4)mod9 = 4
and so on..

A pattern of 4,7,1 will be repeated
4^(3k+1) will leave remainder 4 when divided by 9
4^(3k+2) will leave remainder 7 when divided by 9
4^(3k) will leave remainder 1 when divided by 9

Now 32 = (3*10 + 2))

Therefore, continuing from (i)
= (4^(3k+2)^32)mod9
= (4^32)mod9
=(4^(3k+2))mod9
=4