Question

Find the remainder when 33333.....36 times is divided by 19?

Answer 1

Heya user,
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Clearly, 

33333333333... = 3 * ( 1111111111... 36 times) 
= 3 * [10^(36) - 1] / 9 = [ 10^36 - 1 ] / 3

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Now, Ф(19) = 18
By Euler's phi function, 10^18 ≡ 1 (mod 19)

=> 10^36 ≡ 1 (mod 19)
=> [ 10^36 - 1 ] ≡ 0 (mod 19)

Hence, 19 divides [ 10^36 - 1 ]... and so------>

19 divides ---> [ 10^36 - 1 ] / 3 
=> 19 | 3333333333... { 36 times }

Hence, the remainder is 0.