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How many integers n, 1 <= n <= 4000, are multiples of 16 or 30 or both?

 Jun 29, 2022
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Start with numbers divisible by 16. The smallest number is 16 (16 x 1), and the largest number is 4000 (16 x 250). This makes for 250 integers. 

 

Now, we do the numbers divisible by 30. The smallest number is 30 (30 x 1), and the largest is 3990 (30 x 133). This makes for 133 integers. 

 

However, we double count for every multiple of 240 (lcm of 16 and 30). The smallest number we overcount for is 240 (240 x 1) and the largest is 2840 (240 x 16). This makes for 16 integers. 

 

Thus, there are \(250 + 133 - 16 = \color{brown}\boxed{367}\) integers.

 Jun 29, 2022
edited by BuilderBoi  Jun 29, 2022

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