How many positive integers less than 1000 are divisible by neither 2 nor 5?

**From 1 to 999:** divisible by 2 → 499, by 5 → 199, by 10 → 99. By inclusion-exclusion, divisible by 2 or 5 $= 499+199-99 = 599$. Neither $= 999 - 599 = 400$.