How should I start it?

$\text{Out of numbers}<1000, \\ \Rightarrow \lfloor \frac{999}{2} \rfloor =499 \text{ multiples of } 2. \\ \Rightarrow \lfloor \frac{999}{3} \rfloor=333 \text{ multiples of }3. \\ \Rightarrow \lfloor \frac{999}{5} \rfloor=199 \text{ multiples of }5. \\ \Rightarrow \lfloor \frac{999}{6} \rfloor=166 \text{ multiples of }6. \\ \Rightarrow \lfloor \frac{999}{10} \rfloor=99 \text{ multiples of }10. \\ \Rightarrow \lfloor \frac{999}{15} \rfloor=66 \text{ multiples of }15. \\ \Rightarrow \lfloor \frac{999}{30} \rfloor=33 \text{ multiples of }30. \\ \text{Therefore, } 499+333+199-166-99-66+33=733 \text{ numbers are divisible by at least one of }2,3,5. \\ \text{Out of the remaining }999-733=266 \text{ numbers, }165 \text{ are prime other than }2,3,5. \\ \color{#D61F06}{\text{Note: }1\text{ is neither prime nor composite.}} \\ \text{This leaves us with }\boxed{100} \text{ prime looking numbers.}$