Question

Find the remainder when 1234567891011121314151617181920......424344 is divided by 45?

Answer 1

Solution:

Let N = 1234567891011121314151617181920......424344

When N is divided by 5,  then remainder will be 4.

So, N = 5K + 4         ...(1)

When N is divided by 9, then the sum of the digits of N divided by 9.

Since, 1+2+3+...44 = 990

This gives digit sum as 9. So, when N is divided by 9, then the remainder will be 0.

So, N = 9L         ...(2)

Equation (1) and (2), we 9L = 5K + 4

Let K = 1. then the equation satisfied.

Since least possible number satisfies the condition which is  9.

The general format of N = w(LCM of (9, 5)) + Least number satisfies the condition.

So N = w.45 + 9

When N is divided by 45, we obtain remainder 9.