Try this problem from Duke Math Meet 2009 Problem 7 based on the number of ordered triples. This problem was asked in the team round.
How many ordered triples of integers (a, b, c) are there such that $ 1 \le a, b, c \le 70 $ and $ a^2 + b^2 + c^2 $ is divisible by 28.
Any square quantity is 0 or 1 modulo 4
Now the sum of three square quantities is 0 or 1 mod 4. Since 28 is 0 mod 4, therefore none of the three squares can be 1 mod 4.
Thus a, b, c are even.
Again a square quantity is always 0, 1, 4, 2 mod 7
Since we want the three squares to add up to 0 mod 7 (28 is 0 mod 7), the possibilities are (0, 0, 0) mod 7 and (1, 4, 2) mod 7
So we have the following cases:
Case 1
a, b, c are all even multiples of 7. There are 5 of them. Since we are looking for ordered triplet there are $latex 5^3 = 125 $ of these$
Case 2
a, b, c are all even numbers. with 1, 6 mod 7, 2, 5 mod 7 and 3, 4 mod 7
1 mod 7 - 10 elements
6 mod 7 - 10 elements
Hence there are 20 numbers which are 1 or 6 mod 7. 10 of these are even.
Similarly, there are 20 numbers which are 2 or 5 mod 7 and 20 numbers which are 3 or 4 mod 7. Half of each of which we will take.
So 10 choices for each set. $ 10^3 = 1000$ choices. Now since we are taking ordered pairs, we must consider all the $ 3! = 6 $ permutations. So there are 6000 cases.
Hence answer is $ 6000 + 125 = 6125 $
Publishing original scientific research in a peer-reviewed journal is an achievement many scholars work toward for years. Achieving this feat early in one's academic journey requires exceptional intellectual rigor, creativity, and persistence. Aaditya Punatar and Muskan Sadana have reached this remarkable milestone, successfully publishing their collaborative research paper in the prestigious Journal of Emerging Investigators […]
Her project on quantum-accelerated Steiner trees for 15-minute cities earned a place among the finalists at the world's largest pre-college STEM competition. Prishaa Shrimali of Marymount School of New York, was selected as a Grand Finalist at the 2026 Regeneron International Science and Engineering Fair (ISEF) in Phoenix, Arizona. Her project was Optimizing Urban Accessibility: […]
In 2026, the following Cheenta students have been successful for Indian Statistical Institute's M.Stat Entrance. They ranked within the first 50 in the entire country in these entrances. I.S.I. M.Stat Entrance
In 2026, the following Cheenta students have been successful for Indian Statistical Institute's B.Stat Entrance and Chennai Mathematical Institute's B.Sc. Math Entrance. They ranked within the first 200 in the entire country in these entrances. Most of these students attended the problem solving workshops regularly, which happen 5 days every week. CMI B.Sc. Math Entrance […]
