How many number between 1 and 1000 satisfy a certain condition?

Question

How many positive integers less than $1,000$ are multiples of $5$ and are equal to $3$ times an even integer?

It is simply asking for multiples of $5$ and $6$

Is there a way to do this without guessing or counting? This is from the SAT.

I have tried it, but I counted all, and found common ones, the answer was $33$.

Answer 1

For a number to be a multiple of $x$ and $y$ it must be a multiple of the LCM of $x$ and $y$. So in this case, how many integers less than 1000 are multiples of 30? The 33rd multiple of 30 is 990, the 34th is 1020, so 33 it is.

Answer 2

Note that a number is a multiple of both $m$ and $n$ if and only if it is a multiple of $\text{lcm}(m,n)$. In this case, this is $30$. So the question sis how many positive itnegers are less than $1000$ and divisible by $30$. The answer is $ \left\lfloor \frac {1000}{30} \right\rfloor = 33 $. Can you see why?

Answer 3

If it's a multiple of two numbers it is a multiple of the least common mutliple, in this case the least common multiple of $6$ and $5$ is $30$. You now want to find how many multiples of $30$ are under $1000$

Hint: $33(30)=990,34(30)=1020$

Answer 4

Don't get yourself confused, "multiples of 5 and 6" could mean something else, as in, for example, "multiples of 5 and multiples of 6;" that would be quite something else entirely. Let's look at the exact wording of the question, which is the following, right?

How many positive integers less than 1,000 are multiples of 5 and are equal to 3 times an even integer?

So, multiples of 5 include: 5, 10, 15, 20, 25, 30, 35, ..., 990, 995, 1000. But there's another condition and we need to remove some of these multiples of 5. We need multiples of 5 that are also thrice an even integer. So we need multiples of 10: 10, 20, 30, 40, 50, 55, 60, 65, 70, ..., 990, 1000. But they also need to be multiples of 3, so what we're really looking for are multiples of 30: 30, 60, 90, 120, 150, 180, 210, ..., 990.

Without guessing or counting? You can compute $\frac{1000}{30}$ and round down to an integer, in this case, 33 (notice that $34 \times 30 = 1020 > 1000$).

Now, me, I like to do these things with guessing and counting. Notice that $30^2 = 900$. After 900, there are another three multiples of 30 before we go over 1000, so that's another way to get 33.