2011/016) a problem with repunit

A number is called repuint if all the digits of the number are 1 that is it is 1 , 11 , 111 so on

in a repunit number with 50 1's the 26th digit from the left is modified to some digit so that is is divisible by 13. what is the digit

Ans:
method 1
say the digit x
the number is 1(25 times)x1(24 times)
= (10^50-1)/9 + (x-1)(10^25-1)/9

we need to find x such that
(10^50-1)/9 + (x-1)(10^25-1)/9 mod 13 = 0

as 9 is coprime to 13

so (10^50-1)+ (x-1) )(10^25 -1)mod 13 = 0


as 10 and 13 are coprime so as per format's little theorem


10^12 = 1 mod 13

so 10^48 = 1 mod 13

so 10^50 = 100 mod 13

so (10^50-1)+ (x-1) )(10 -1) =

(100-1) + 9(x-1) mod 13 = 0

99 + 9x - 9 or 90+9x = 0 mod 13 or 10 +x = 0 mod 13 or x= 3

so ans = 3

Method 2


sequence of 12 1's is divisible by 13 so we subtract sequence of 12 1's and divide by 10^12 and do it 2 times

we get 1...(25times)x

as seqeunece of sequence of 12 1's is divisible by 13 so multiplying by any number so multiply by 10^14 and subtract

we get 1..(13 times) x

as sequence of 12 1's is divisible by 13 so multiplying by any number so multiply by 10^2 and subtract

1x is the number and divisible by 13 so x = 3