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Question Number 193449 by Mingma last updated on 14/Jun/23

Commented by Frix last updated on 14/Jun/23

$$\mathrm{The}\mathrm{remainder}\mathrm{of}{10}^n-1\mathrm{divided}\mathrm{by}625$$$$\mathrm{is}624\mathrm{for}n⩾4$$

Commented by Mingma last updated on 14/Jun/23

Can you explain in details?

Commented by Frix last updated on 14/Jun/23

$$n⩾4:$$$${10}^n-1={10}^n-625+624=$$$$={10}^{n-4}\times 10000-625+624=$$$$={10}^{n-4}\times 16\times 625-625+624=$$$$=({10}^{n-4}\times 16-1)\times 625+624$$$$\iff$$$$\frac{{10}^n-1}{625}={10}^{n-4}\times 16-1+\frac{624}{625}$$

Commented by Mingma last updated on 14/Jun/23

Perfect ��

Answered by MM42 last updated on 14/Jun/23

$$625=5^4$$$$N={10}^{100}-1=2^{100}\times {\left(5^4\right)}^{25}-1\stackrel{625}{\equiv }-1\stackrel{625}{\equiv }624✓$$

Commented by Mingma last updated on 14/Jun/23

Perfect ��

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