Need proof (1)

It must have 2 numbers among 7 , 77 , 777 , 7777 , ... , 777...7
.......................................................................................(m+1 7s)
have the same remainder when they are divided by m since there are only
m remainders.
Let 777...7 and 777...7
......(a 7s)............. (b 7s) (a > b)
are 2 numbers have the same remainder when they are divided by m.
then
777...7 - 777...7
(a 7s) ...... (b 7s)

= 777... 7 000...0 is a multiple of n.
..(a -b 7s) (b 0s)