How many whole numbers from 1 to 100 leave remainder 2 on division by 3 and remainder 3 on division by 5?

The two conditions first hold together at $8(8=3\times 2+2$ and $8=5\times 1+3$), and then repeat every LCM(3, 5) = 15. The numbers are 8, 23, 38, 53, 68, 83 and 98 — the next, 113, is beyond 100. So there are 7 of them.