I know that the number of integers between 1 and 1000 that are divisible by 30
is 33, and the number of integers between 1 and 1000 that are divisible by 16 is 62, but I do not know how to calculate the number of integers that is divisible by both 30 and 16, especially because they are not primes. Could anyone help me please?
[Math] Number of integers between 1 and 1000 that are divisible by 30 but not divisible by 16.
divisibilityelementary-number-theorygre-exam
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Best Answer
It does not matter if they are primes.
So here we are trying to calculate the number of multiples of $30$ which are not multiples of $16$ below thousand. For this, we take all multiples of $30$ below thousand first, and then subtract from this, all numbers that are multiples of $30$ and $16$ which are below thousand.
Any number is a multiple of $30$ and $16$ if and only if it is a multiple of their least common multiple, which in our case is $240$, which I computed by prime factorization, if you wanted to know how that is done.
So the answer is the number of multiples of $30$ minus the number of multiples of $240$ which are less than thousand. This is then $33-4 = 29$.