Not less than 60 days before the election, any 10 or more voters in any division of the district may file with the board a petition requesting that the name of a specified person be placed on the ballot for the particular division as a candidate for the office of director to represent the particular division. Only the names proposed by the petitions filed with the board shall be printed on the ballots for the various divisions of the district. The date of actual receipt of any such petition by the board shall be deemed to be the date of filing of the same with the board. If there are less than 10 voters in any division, signatures of one-third of the number of voters, rounded up to the next whole number, shall be required to nominate a candidate.