JMO - Division

If S = 123456789101112................9979989991000

What will be the Remainder when S is divided by 9.




Answer:

first take the single digit number. each of the digits from 1 to 9 occur once. sum = 9 x 10 /2
(note the sum is a multiple of 9)
now take the 2 digit numbers, there are (99 - 10 + 1 = 90) numbers and  180 digits. Each digit from 1 to 9 occurs equal number of times.
sum = 20 x (9 x 10/2) ......because 180/9 = 20. so each number makes 20 appearance in all two digit numbers.
(note the sum is a multiple of 9)
now consider the 3 digit numbers, there are (999 - 100 + 1 = 900) numbers with 2700 digits. Each digit from 1 to 9 occurs equal number of times.
Sum = 300 x (9 x 10/2)........because 2700/9 = 300. so each number makes 300 appearance in all three digit numbers.
(note the sum is a multiple of 9)
SInce all the sums upto 3 digit number 999 are a multiple of 9, the Remainder is zero.
the only number left is 1000. 1000/9 = 111 (1/9)
So, the Remainder is 1.