Define the squences a_1, a_2, ... and b_1, b_2, ... by a_1 = b_1 = 7 and a_n+1 = (a_n)^7, b_n+1 = 7^(b_n)
Find the last digit of a_2009 and of b_2009
What about the last two digits? or more?
Solution for finding out the last digits:
its clear that
since ,
we have
so
so
so finally
so .
The last digit of is 7.
for observe that since , for each .
now .
write for some integer .
so
compute that ,
so that .
convince yourself that for each and hence prove that is odd.
this will lead to the result that The last digit of is 3.