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How many positive integers less than 2020 aren’t divisible by 2, 3, 5, or 7?

 Jan 17, 2021
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We can use PIE and complementary counting. The numbers that ARE divisible by 2,3,5,7 are going to be (divisible by 2)+(divisible by 3)+(divisible by 5)+(divisible by 7)-(divisible by 2*3)-(divisible by 2*5)-(divisible by 2*7)-(divisible by 3*5)-(divisible by 3*7)-(divisible by 5*7)+(divisible by 2*3*5)+(divisible by 2*3*7)+(divisible by 2*5*7)+(divisible by 3*5*7)-(divisible by 2*3*5*7). This yields 1010+673+404+288-336-202-144-134-96-57+67+48+28+19-9=1559

:D

 Jan 17, 2021

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