Number of integers between 1 and 1000 that are divisible by 30 but not divisible by 16. 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?

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 $33-4 = 29$.

$$N=2×3×5k=2^4K$$

$$\implies K=3×5K'$$ by Gauss Theorem.

thus $N$ is a multiple of $$3×5×16=240$$

hence $$N=240,480,720,960$$

• your answer is not clear for me at all, could u provide some more details please? – Intuition Aug 1 '17 at 10:04

Eliminate E. Notice that $90=3\times30$. Eliminate D. We know that if we multiple $90$ by $2$ we have $180$ which is divisible by $30$ but not $16$. Eliminate C. Multiply $180$ by $2$ and we have $360$ is divisible by $30$ but not by $16$, so eliminate B. The answer is A.

• your answer is not clear for me at all. – Intuition Aug 1 '17 at 9:47
• We can eliminate E right away because in your original post you said only 33 integers between 1 and 1000 are divisible by 30. Next we can eliminate D because we know 90 is divisible by 30 but not 16. After that we can also eliminate C because we know 180 is divisible by 30 but not 16. Then we can eliminate B because we know that 360 is divisible by 30 but not by 16. – JohnColtraneisJC Aug 1 '17 at 16:16