13.

A box of hawthorns has more than one hundred grains. When counted in groups of 3, there is 1 extra; when counted in groups of 4, there are 2 extra; when counted in groups of 5, there are 3 extra; and when counted in groups of 6, there are 4 extra. The box of hawthorns has at most ( ) grains.
【Analysis】A number over one hundred, when divided by 3 leaves a remainder of 1, when divided by 4 leaves a remainder of 2, when divided by 5 leaves a remainder of 3, and when divided by 6 leaves a remainder of 4. Therefore, this number plus 2 is a common multiple of 3, 4, 5, and 6, and this number is of the form $60k-2$. Considering it is over one hundred, the maximum is 178