The specific failure was that I first attempted to solve by using repeated subtraction. The guy kept asking me to "solve it a different way", or saying "there is a better way to solve this". I tried using arithmetic tables, I tried using results about base10 remainders and I even tried using one of the corollaries to Fermat's little theorem to speed it up for larger inputs... every time I was told I was getting it wrong because "there was a better solution". In the end he pointed out that the only solution he would accept was use of the mod operator.
Since then I have actively kept a tally: I naturally use the mod operator an average of twice a calendar year, it has always been in personal life code when dealing with complicated geometry problems, the bit of code containing it almost always fails on some edgecase because at the point of using mod it is convoluted.