9×109....989

 

n是不小於2的正整數,令a=10×10n+(10n-21)×100+89,則a的反序數 b=98×10n+(10n-21)×100+1,

a+b=108×10n+(10n-2-1)×200+90=108×10n+(10n-2+10n-2-2)×100+90=109×10n+(10n-2-2)×100+90.

而10a=1010n+(10n-21)×1000+890=109×10n+(10n-21)×10009×10n+890=

109×10n+10n+110009×10n+890=109×10n+10n1000+890=109×10n+10n200+90=
109×10n+(10n-2-2)×100+90= a+b.

因此 9a=b.

例如:[註1]

n=2時,9×1089=9801.

n=3時,9×10989=98901.

n=4時,9×109989=989901.

n=5時,9×1099989=9899901.

.............

 

[註1]: ggb檔


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