N = 7777..................7777, where 7 repeats itself 603 times. What is the remainder left when N is divided by 1144?

The answer is 777

1144 = 8*13*11
Now 77777..... mod 11 will be 7 only as divisibility of 11 is : (sum of digits at odd places- sum of digits at even places )
so we get 11k+7
Now 777777...... 777 mod 8 will be 777 mod 8 ( divisibility of 8 is last 3 digits ) we get 777mod 8 =1 so we get 8n+1
Now 77777777777 mod 13 now Euler (13) = 12 so 77777777.. 600 times mod 13=0 and 777777.... 603 times mod 13 will be 777 mod 13 = 10
so we get
11k+7 =8n+1 =13m+10
Simultaneously solving them
we get k=70 , n =97 , m=59 and remainder as 777.