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Let S be a subset of {1, 2, 3, • • • , 2021} such that no pair of distinct elements in S has a sum divisible by 7. What is the maximum number of elements in S? 

 Aug 22, 2022
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The set would contain 1 number that is divisible by 7, and as many "sets" of numbers that leave a remainder of 1, 2, and 3 when divided by 7. 

 

The smallest "set" of numbers would be 1, 2, and 3, and the largest "set" of numbers would be 2017, 2018, and 2019. 

 

The smallest number is \(7 \times 0 + 1\) and the largest number is \(7 \times 288 + 1 \), so there are \(288 - 0 + 1 = 289\) "sets" of numbers. 

 

So, set S can have at most \(289 \times 3 + 1 = \color{brown}\boxed{868}\) elements.  

 Aug 23, 2022

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