Question

Find the remainder when 7777 .... (upto 37 digits) is divided by 19.

Answer 1

Heya user, 
just answered similar question, so I'll just copy paste the main part...

7777777... = 7 * ( 1111111111... 36 times) 
= 7 * [10^(36) - 1] / 9
_____________________________________________________________

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 ]

____________________________________________________________

Again, 7777777777777... ( 37 times ) = 10[ 777... ( 36 times ) ] + 7

We see, 19 divides [ 10^36 - 1 ], and so, 
19 | ( 1111111... { 36 times } )

Hence, 19 divides 10 * 7 [ 11111111111.. {36 times} ]

But,
7777777777777... ( 37 times ) = 10[ 777... ( 36 times ) ] + 7 = 19x + 7

Hence, rem. when 777... { 37 times } is divided by 19 is 7...