Answer
10 integers
Work Step by Step
For the numbers between 100 and 999, there are only 9 numbers that share no equivalent digits in them: 111,222,333,444,555,666,777,888, and 999. A total of 9 numbers. Therefore, if we pick one more number, it would have to start with one of the previous integers chosen (therefore sharing a common digit). So we would have to choose a minimum 10 digits to assure this.