None of the grounds of contest enumerated in Section 17-13-71 shall serve to annul or set aside any declared nomination, unless such ground alone or in conjunction with other of such grounds alleged shall serve to show to the reasonable satisfaction of the committee trying the contest that the person who was declared nominated and whose nomination is contested did not receive at such primary the number of legal votes necessary under this chapter to be nominated. But upon such contest, the committee shall have power to declare who was legally nominated at such primary for such office.