Sum of first 1000 positive integers: (S=\dfrac{1000\cdot1001}{2}=500500)

Sum of multiples of 2: (2+4+\cdots+1000=\dfrac{500}{2}(2+1000)=250\cdot1002=250500)

Sum of multiples of 5: (5+10+\cdots+1000=\dfrac{200}{2}(5+1000)=100\cdot1005=100500)

Sum of multiples of 10 (both): (10+20+\cdots+1000=\dfrac{100}{2}(10+1000)=50\cdot1010=50500)

Sum of integers divisible by 2 or 5: (250500+100500-50500=300500)

Required sum: (500500-300500=200000)

Answer: (200000)