The large number has been formed by concatenating the integers 1 to 44. What is the remainder when it is divided by 45?

Question

The large number (123456...424344) has been formed by concatenating the integers 1 to 44. What is the remainder when it is divided by 45?

Note: solve without calculation...

Answer 1

9 is the remainder
The number on dividing by 5 leaves remainder 4.
Digitsum of Numbers (1234......424344)-->
The number are from 0 to 9 (Sum of digits 45)
then 10 to 19 (Digit sum 45+1x10 = 55)
then 20 to 29 (Digit sum 45+2x10 = 65)
then 30 to 39 (Digit sum 45+3x10 = 75) and
then 40 to 44 (digit sum (10+4x5 = 30)
therefore total digit sum come out to be (45+55+65+75+30 = 270)
This is divisible by 9.
When the given number is divided by 5 it leaves a remainder 4. So the number is of the form 5A + 4.
When divided by 9 it leaves a remainder 0. Hence of the form 9B+0.
Equate both the equations 5A+4=9B.
Put A= 1 and B= 1 we get 9 which is the remainder.