问题及解答

证明:

\[
13\underbrace{88\cdots 8}_{6k}91\equiv 0(\mod 13),\quad\forall\ k=0,1,2,\ldots
\]

且

\[
13\underbrace{88\cdots 8}_{72k}91\equiv 0(\mod 169),
\]

即第一个能被169整除的是 1388888888888888888888888888888888888888888888888888888888888888888888888891， 其中含有72个8.

以上将所有8改成任何数字都一样. 这是因为 xxxxxx00 始终能被 13 整除.

1388888888888888888888888888888888888888888888888888888888888888888888888891 = 13 * 13 * 7696738667 * 6838252872218262661 * 156145294634054688436224492369146997125414197

一些计算（使用的是 ARIBAS 软件）

==> 1388888891 mod 13.
-: 0

==> 1388888891 mod 169.
-: 78

==> 1388888888888891 mod 13.
-: 0

==> 1388888888888891 mod 169.
-: 117

==> 1388888888888888888891 mod 13.
-: 0

==> 1388888888888888888891 mod 169.
-: 156

==> 1388888888888888888888888891 mod 13.
-: 0

==> 1388888888888888888888888891 mod 169.
-: 26

==> 1388888888888888888888888888888891 mod 13.
-: 0

==> 1388888888888888888888888888888891 mod 169.
-: 65

==> 1388888888888888888888888888888888888891 mod 13.
-: 0

==> 1388888888888888888888888888888888888891 mod 169.
-: 104

==> 1388888888888888888888888888888888888888888891 mod 13.
-: 0

==> 1388888888888888888888888888888888888888888891 mod 169.
-: 143

==> 1388888888888888888888888888888888888888888888888891 mod 13.
-: 0

==> 1388888888888888888888888888888888888888888888888891 mod 169.
-: 13

==> 1388888888888888888888888888888888888888888888888888888891 mod 13.
-: 0

==> 1388888888888888888888888888888888888888888888888888888891 mod 169.
-: 52

==> 1388888888888888888888888888888888888888888888888888888888888891 mod 13.
-: 0

==> 1388888888888888888888888888888888888888888888888888888888888891 mod 169.
-: 91

==> 1388888888888888888888888888888888888888888888888888888888888888888891 mod 13.
-: 0

==> 1388888888888888888888888888888888888888888888888888888888888888888891 mod 169.
-: 130

==> 1388888888888888888888888888888888888888888888888888888888888888888888888891 mod 13.
-: 0

==> 1388888888888888888888888888888888888888888888888888888888888888888888888891 mod 169.
-: 0

==>