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.
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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.
