Find the smallest positive integer multiple of 84 whose base ten representation uses only the two digits 6 and 7.

The following is a table of 6*10^n and 7*10^n mod 84:
A      B
6      7
60    70
12    28
36    28
24    28
72    28
48    28

Pick one from column A and one from column B, sum these and look for numbers that are 0 mod 84.
There is no way to get this result with any number less than 5 digits.
An obvious choice is 77700 since 3*28 = 84.
But 76776 is smaller and works:  28+36+28+70+6 = 168 = 2*84

Posted by Larry on 2014-08-02 15:17:38