Find the smallest positive integer that is a multiple of 35 whose digits are all the same.

The prime factors of 35 are 5 and 7. As any number that has a factor of 5 must end in 5 or 0, a number that is both a multiple of 35 and have all the same digits must then have all fives.

The smallest is with 6 fives: 555,555.
The prime factors: 3 * 5 * 11 * 13 * 37.

The following are the next three multiples:

12 fives: 555,555,555,555 = 3 * 5 * 7 * 11 * 13 * 37 * 101 * 9901.

18 fives: 555,555,555,555,555,555 = 3² * 5 * 7 * 11 * 13 * 19 * 37 * 52579 * 333667.

24 fives: 555,555,555,555,555,555,555,555 = 3 * 5 * 7 * 11 * 13 * 37 * 73 * 101 * 137 * 9901 * 99990001.

Posted by Dej Mar on 2006-05-03 13:16:24