o.O So for any such qn, we can use this method? If suppose the digits were not repeating, then also same method?

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If $N = 888 $ up to $100$ digits, what is the remainder when $N$ is divided by $625?$

- 128
- 138
- 338
- 388

+4 votes

Best answer

$625 \times 16 = 10000$

i.e. anything which is multiple of 10000 will also be multiple of 625

$\underbrace{8888 \cdots 8}_{\text{100 times}} =\underbrace{8888 \cdots 8}_{\text{96 times}} \times 10000 +8888$

So to find reminder of $\underbrace{88888888888888\cdots 8}_{\text{100 times}}$, just find Reminder when 8888 is divided by 625

$8888 = 625 \times 14 + 138$

So, 138 is reminder.

i.e. anything which is multiple of 10000 will also be multiple of 625

$\underbrace{8888 \cdots 8}_{\text{100 times}} =\underbrace{8888 \cdots 8}_{\text{96 times}} \times 10000 +8888$

So to find reminder of $\underbrace{88888888888888\cdots 8}_{\text{100 times}}$, just find Reminder when 8888 is divided by 625

$8888 = 625 \times 14 + 138$

So, 138 is reminder.

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