quiz problem of positive integers

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  • #29470

    Aradhya Goel
    Participant

    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.

    #29506

    Nitin Prasad
    Participant

    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!

    • This reply was modified 2 weeks, 3 days ago by  Nitin Prasad.
    • This reply was modified 2 weeks, 3 days ago by  Nitin Prasad.
    • This reply was modified 2 weeks, 3 days ago by  Nitin Prasad.
    • This reply was modified 2 weeks, 3 days ago by  Nitin Prasad.
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