When the digit is placed at the beginning and end of a positive integer N, the new number is 99*N.

Find the smallest value of N.

**Let
as start by assuming that the digit is 1**

**N=
(10^k+1)/89**

**It
means that 10^k should be 88 mod 89**

**100=11
mod 89 , 1000=110 mod 89=21 ****mod 89 ****, 1000=32 mod 89… later,
53, 85,..etc....80,88**

**Continuing
this process with a simple calculator we arrive at the correct value, i.e, k=22, meaning that with digit 1 our 89*N**

is 1000….00 (21 zeroes)0… ..1, and N derived by dividing this number by
89 is

__N=112,359,550,561,797,752,809__

**Checking with
scientific calculator:**

**99*112,359,550,561,797,752,809
=11,123,595,505,617,977,528,091 ok**

**Since 89
is a prime number, there is clearly no way that digit other than 1 could bring
a lesser result for N.**