Question

A hundred one digits number is formed by writing natural numbers (starting from 1) as 12345678910111213…. Find the remainder when this number is divided by 8.

Answer 1

Since $8\mid 10^3$, it is sufficient to check the last $3$ digits.

1-9 : 9 digits
10-55 : 46*2 = 92 digits

That means the number ends with $\ldots 53545$ and the last 3 digits are $545$ :
$545=8\times68+\boxed{1}$