Question Number 193449 by Mingma last updated on 14/Jun/23
Commented by Frix last updated on 14/Jun/23
$$\mathrm{The}\:\mathrm{remainder}\:\mathrm{of}\:\mathrm{10}^{{n}} −\mathrm{1}\:\mathrm{divided}\:\mathrm{by}\:\mathrm{625} \\ $$$$\mathrm{is}\:\mathrm{624}\:\mathrm{for}\:{n}\geqslant\mathrm{4} \\ $$
Commented by Mingma last updated on 14/Jun/23
Can you explain in details?
Commented by Frix last updated on 14/Jun/23
$${n}\geqslant\mathrm{4}: \\ $$$$\mathrm{10}^{{n}} −\mathrm{1}=\mathrm{10}^{{n}} −\mathrm{625}+\mathrm{624}= \\ $$$$=\mathrm{10}^{{n}−\mathrm{4}} ×\mathrm{10000}−\mathrm{625}+\mathrm{624}= \\ $$$$=\mathrm{10}^{{n}−\mathrm{4}} ×\mathrm{16}×\mathrm{625}−\mathrm{625}+\mathrm{624}= \\ $$$$=\left(\mathrm{10}^{{n}−\mathrm{4}} ×\mathrm{16}−\mathrm{1}\right)×\mathrm{625}+\mathrm{624} \\ $$$$\Leftrightarrow \\ $$$$\frac{\mathrm{10}^{{n}} −\mathrm{1}}{\mathrm{625}}=\mathrm{10}^{{n}−\mathrm{4}} ×\mathrm{16}−\mathrm{1}+\frac{\mathrm{624}}{\mathrm{625}} \\ $$
Commented by Mingma last updated on 14/Jun/23
Perfect ��
Answered by MM42 last updated on 14/Jun/23
$$\mathrm{625}=\mathrm{5}^{\mathrm{4}} \\ $$$${N}=\mathrm{10}^{\mathrm{100}} −\mathrm{1}=\mathrm{2}^{\mathrm{100}} ×\left(\mathrm{5}^{\mathrm{4}} \right)^{\mathrm{25}} −\mathrm{1}\:\overset{\mathrm{625}} {\equiv}\:−\mathrm{1}\overset{\mathrm{625}} {\equiv}\:\mathrm{624}\:\checkmark\: \\ $$
Commented by Mingma last updated on 14/Jun/23
Perfect ��