Puzzle #215: Divvying Up Candy

Six kids were given a bag of candy and told to split it evenly. When they tried to do so, there was one piece of candy left over. There would have been one piece of candy left over if there were only two, three, four, or five kids, too. What is the minimum number of pieces of candy there could be in the bag if there were at least ten pieces of candy?

Submit your answer to Gene Wirchenko <genew@telus.net>. Your answer should be in the form of a proof. That means to show how your answer must be correct. The deadline is Wednesday, July 19, 2017 at noon Pacific Time. I will post the answer shortly after.