Finding the number of multiples

Question

I have recently been doing problem solving in math, and I came across this problem:

Determine the number of positive multiples of $6$ or $9$ or both, less than $1000$.

I appreciate any help. Thanks!

Answer 1

The number of positive multiples of $6$ that are less than $1000$ is $$\left\lfloor\frac{1000}{6}\right\rfloor=166$$ Likewise, the number of positive multiples of $9$ that are less than $1000$ is $$\left\lfloor\frac{1000}{9}\right\rfloor=111$$ Adding these, we get $277$, but we have counted twice the numbers that are multiples of both $6$ and $9$. So we need to count those numbers, then subtract from the total. Numbers that are multiples of both $6$ and $9$ are numbers that are multiples of the least common multiple of $6$ and $9$. That least common multiple is $18$, so we find: $$\left\lfloor\frac{1000}{18}\right\rfloor=55$$ Then subtracting from $277$ we have the final count: $$277-55=222$$