2021 AMC 10 B Fall
Complete problem set with solutions and individual problem pages
For each integer , let be the sum of all products , where and are integers and . What is the sum of the 10 least values of such that is divisible by 3 ?(2021 AMC Fall 10B, Question #22)
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- B.
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Solution 1:
To get from to , we add Now, we can look at the different values of . For and , then we have . However, for , we have Clearly, . Using the above result, we have , and , and are all divisible by 3 . After , we have , and all divisible by 3 , as well as , and . Thus, our answer is
Solution 2:
Since we have a wonky function, we start by trying out some small cases and see what happens. If is 1 and is 2 , then there is once case. We have for this case. If is 3 , we have which is still . If is 4 , we have to add which is a multiple of 3 , meaning that we are still at 2 mod 3 . If we try a few more cases, we find that when is 8 , we get a multiple of 3 . When is 9 , we are adding , and therefore, we are still at a multiple of 3 .
When is 10 , then we get which is 10 times a multiple of 3 . Therefore, we have another multiple of 3 . When is 11 , so we have 2 mod 3 . So, every time we have , and , we always have a multiple of 3 . Think about it: When is 1 , it will have to be , so it is a multiple of 3 . Therefore, our numbers are . Adding the numbers up, we get (B) 197
