between 1 and 2009, how many numbers are multiples of 5 and 7

Between 1 and 2009, how many numbers are multiples of 5 and 7?

Since the numbers must be multiples of 5 AND 7, it would be easier to just find multiples of 7 that end in 5 or 0. For example, 7*5 = 35, 7*10 = 70, 7*15 = 105, . . 140, 175, 210, 245, 280, 315, 350, 385, 420. . .  

Notice the pattern: 7 is simply being multiplied by multiples of 5. So a faster way to find the number of multiples of 5 and 7 from 1 - 2009 would be to divide 2009 by 7, = 287. The closest multiple of 5 under 287 is 285.

Then, divide 285 by 5, = 57. So there are 57 multiples of both 5 and 7 between 1 and 2009.

- - - - - - - - - - - - - - - - - -  - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

Actually, there is a much simpler way (oops):

The least common multiple of 5 and 7 is 35. Divide 2009 by 35, = 57.4

Rounded down, that is 57.

So this way also results in 57 multiples of 5 and 7 between 1 and 2009.

I interpreted it differently from you Kitty but when I read the question again your interpretation is the correct one.

You found the intersection I was going to find the union. 

$$\begin{array}{rcr} \textcolor[rgb]{1,0,0}{1}\times5\times7=1*35&=&35\\ \textcolor[rgb]{1,0,0}{2}\times5\times7=2*35&=&70\\ \textcolor[rgb]{1,0,0}{3}\times5\times7=3*35&=&105\\ \textcolor[rgb]{1,0,0}{4}\times5\times7=4*35&=&140\\ \textcolor[rgb]{1,0,0}{5}\times5\times7=5*35&=&175\\ ...&=&...\\ \textcolor[rgb]{1,0,0}{n}\times5\times7=n*35&\approx&2009\\ \end{array}$$

$$n\approx\dfrac{2009}{5\times7}=\dfrac{2009}{35}=57.4$$

 Between 1 and 2009, how many numbers are multiples of 5 and 7:

$$\boxed{n=57}$$