Answer to Qn b)

 

Fermat's little theorem gives 2¹⁶-1 is divisible by 17 as 17 is a prime.

 

Hence 2¹⁶ = 17k+1

Now 2006 = 16.125+6

Hence 2²⁰⁰⁶ = (2¹⁶)¹²⁵*2⁶ = (17k+1)¹²⁵*64

 

= (17n+1)64 = 17m+64 = 17m+17*3+13

 

Hence the remainder is 13

The approach is similar to Feynmann's but the topic of congruences may not be familiar to all. The binomial theorem approach is easier to understand.

Knowing number theory up to Fermat's little theorem is unavoidable however.