Question

What is the number of positive integers less than or equal to 1000 that are divisible by neither 2 nor 5?

• A. 400
• B. 300
• C. 500
• D. 600

Hint:

Use Inclusion-Exclusion principle: Add separate counts, subtract intersection (common multiples).

Answer 1

The Correct Option is A

We want numbers \( \leq 1000 \) not divisible by 2 or 5 Total numbers = 1000 Numbers divisible by 2 = \( \left\lfloor \frac{1000}{2} \right\rfloor = 500 \)
Numbers divisible by 5 = \( \left\lfloor \frac{1000}{5} \right\rfloor = 200 \)
Numbers divisible by both 2 and 5 = divisible by 10 = \( \left\lfloor \frac{1000}{10} \right\rfloor = 100 \)
By inclusion-exclusion: \[ \text{Div by 2 or 5} = 500 + 200 - 100 = 600 \Rightarrow \text{Not divisible by 2 or 5} = 1000 - 600 = \boxed{400} \]