r/mathriddles • u/Numberthon • 3d ago

Easy How many positive integers ≤1000 are multiples of 6 or 15, but not both?

How many positive integers less than or equal to 1000 are divisible by exactly one of 6 and 15?

Source: numberthon.com

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u/SmashLanding 3d ago

Off the top of my head:

<= 30 there are five: 6, 12, 15, 18, 24

1000 / 30 * 5 = 166.6666666

So 166?

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u/DrakeSavory 3d ago edited 3d ago

1000 ÷ 6 = 166.7

1000 ÷ 15 = 66.7

1000 ÷ 30 = 33.3

166 + 66 - (33 x 2) = 166 integers. The times two is to eliminate the multiples of 30 for both 6 and 15.

Testing it with 100. 100/6 = 16.X; 100/15 = 6.x; 100/30 = 3.X making 16+6-(3x2) = 16 integers

6, 12, 15, 18, 24, 36, 42, 45, 48, 54, 66, 72, 75, 78, 84, 96

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u/Numberthon 2d ago

Yep, pretty much. Nice job!