quiz problem of positive integers

[Aradhya Goel]

the question is :

The number of positive integers less than or equal to 10,000 which are divisible by neither 3 nor 5 is
Select one:
a.3332
b.3665
c.2666
d.2999

the answer given is (c)

my answer is coming as 5333.

[Nitin Prasad]

Let-

1. $$A=\{ n : n<10,000 \& n\in \mathbb{N}\}$$
2. $$A_3=\{ n :n \in A \& 3|n\}$$
3. $$A_5=\{ n :n \in A \& 5|n\}$$

Hence, number of positive integers less than or equal to 10,000 which are divisible by neither 3 nor 5

$$=|A_3^c\bigcap A_5^c|$$

$$=|(A_3\bigcup A_5)^c|$$

$$=10,000-|(A_3\bigcup A_5)|$$

$$=10,000-|(A_3|-|A_5)|+|(A_3\bigcap A_5)|$$

$$=10000-\lfloor\frac{10000}{3}\rfloor-\lfloor\frac{10000}{5}\rfloor+\lfloor\frac{10000}{15}\rfloor=10000-3333-2000+666=5333$$

Yeah, your answer is correct!

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