Ports 1, 4, 9, 16, 25, 36,49, 64, 81 and 100 will remain open.
The ports will only remain open if they are toggled an odd number of times, and only if the leg's number has an odd number of factors. Factors have to come in pairs, since you need to multiply two factors to make a number. Square numbers are also the only numbers where there's only one factor involved (multiplying by itself, so it counts as one).
Consider the number 16: 1 x 16, 2 x 8, 4 x 4. Its factors are therefore 1 and 16, 2 and 8, and 4, which makes a total of five factors, an odd number.
Square numbers are the only numbers with an odd number of factors and therefore only those corresponding exhaust ports will remain open after one hundred strikes from the Apple-Crab.