Simple divison!!!!

The remainder that you get when you divide a number by n is called that number modulo n. For example, 23 modulo 4 is 3, since when you divide 4 into 23, you get a quotient of 5 and a remainder of 3.

Note that 45 is the product of 5 and 9, two relatively prime numbers. (Note: two numbers are relatively prime if they have no common factor other than 1.) If you know the number modulo 5 and the number modulo 9, then you can determine the number modulo 45.

The number ends with a 1, so modulo 5 it's 1.

Add all the digits together to get some number, then add all its digits together and continue this to get a digit. In this case its 9, so modulo 9 is 0.

Thus, modulo 5 it's 1, and modulo 9 it's 0. the number between 0 and 45 having this property is the answer.

For a number to be divisible by $45$ , it must be divisible by $5 , 9$ .

Till $123....35$ , the number is divisible by $5$ because last digit is $5$ and sum of digits is also divisible by $9 \quad \quad \quad$ ( $\color{#3D99F6}{\text{To calculate the sum of digits use the formula}} \dfrac{n(n+1)}{2}$ ).

Leftover $3637...4481$ , you can easily divide by $45$ and obtain $\boxed{36}$ as the remainder.