# mod

find tens digit of 3^2011

Note by Alan Liang
6 years, 2 months ago

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Since you can find tens digit of a number n by the result of n mod 100, you can use Euler's Theorem. Thanks to this theorem you know that 3^40 = 1 (mod 100), since φ(100) is 40. Therefore 3^2000 = 1 (mod 100) and 3^2011 = 3^11 (mod 100). 3^11 is 177147 and so 177147 mod 100 = 47. Now you know that 3^2011 mod 100 = 47, and then 3^2011 ends with 4 and 7. 4 is the tens digit.

- 6 years, 2 months ago

Find a pattern.$$03,09,27,81,43,29,87,61,83,49,47,41,23,69,07,21,63,89,67,01,03........$$ These are the last 2 digits for the numbers $$3^{1}$$ to $$3^{21}$$. As you can see,after every 20 numbers the last two digits go back to $$03$$.Hence $$2011/20=100 (r) 11$$.Then we find the 11th number in the 20 number set which is 47,so tens digit of $$3^{2011}$$ is 4.

- 6 years, 2 months ago

Did you learn this from Mathematical Circle? Just curious cause i did so :P

- 6 years, 2 months ago

@Soham What is mathematical circle ?

- 6 years, 2 months ago

First..as your replying to me I take it for granted it's directed towards me .So you didn't have to write "@Soham" :D Anyways itsa legen-waitforit-dary. :D

- 6 years, 2 months ago

A book.

- 6 years, 2 months ago

No,I didn't. I learned it from my olympic maths books :D

- 6 years, 2 months ago

whats that?

- 6 years, 2 months ago

Are you asking what is olympic maths?Olympic Maths is basically a harder version of maths,and uses other methods to solve problems.

- 6 years, 2 months ago