Problem 23 · 2018 Math Kangaroo
Stretch
Number Theory
sum-constraint
Nick wants to split the numbers 2, 3, 4, 5, 6, 7, 8, 9, 10 into some groups so that the sum of the numbers in each group is the same. What is the biggest number of groups he can make this way?
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Answer: B — 3
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Hint 1 of 2
Add 2 through 10 first; each group's sum must divide that total evenly.
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Hint 2 of 2
The number 10 must sit in some group, so every group sum is at least 10 - that caps how many groups you can have.
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Approach: use the total and the largest element to bound the groups
- The numbers 2 + 3 + ... + 10 add to 54.
- Equal groups means each group's sum divides 54, and since 10 sits in one group every group sum is at least 10.
- The only divisor of 54 that is at least 10 and gives more than two groups is 18, which makes 54 / 18 = 3 groups (e.g. {10,8}, {9,7,2}, {6,5,4,3}).
- So the most groups is 3.
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