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?

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- March 23rd 2011, 10:02 PMahm0605find the last two digits
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? - March 25th 2011, 08:26 AMabhishekkgp
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**.