If $N = \underbrace{8\ldots8}_{100 \text{ times} }$, what is the remainder when $N$ is divided by $625$?

Question

If $N = \underbrace{8\ldots8}_{100 \text{ times} }$, what is the remainder when $N$ is divided by $625$ ?

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Is there a method or a particular approach for solving such questions ?

Answer 1

Notice that $10^4=5^4\times 2^4=625 \times 2^4$.

$$N=\underbrace{8\ldots8}_{96 \text{ times} } \times 10^4+8888$$

Hence the remainder when $N$ is divided by $625$ is equal to the remainder when $8888$ is divided by $625$.

Answer 2

$888$ up to $100$ digits?
Do you mean $8\sum_{i=0}^{99} 10^i$?

That would be a $100$ digit number with every digit an $8.$ Putting your number in this form also suggests a solution since 625 divides $10,000.$

Which suggests $625$ divides $8\sum_{i=4}^{99} 10^i$

And then you just need to find the remainder of $8,888/625$