Problem 16 · 2021 Math Kangaroo
Stretch
Logic & Word Problems
divisibilitycasework
In a team competition, there are 5 teams waiting to start. Each team consists of either only boys or only girls. The numbers of team members are 9, 15, 17, 19 and 21. After all members of the first team have started, the number of girls not yet started is 3 times the number of boys not yet started. How many members are on the team that has already started?
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Answer: E — 21
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Hint 1 of 2
After one team leaves, the remaining members split as boys and 3×boys, so the leftover total must be divisible by 4.
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Hint 2 of 2
Total is 81; test which starting team size leaves a multiple of 4 that also splits along whole teams.
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Approach: use the divisible-by-4 leftover and check team splits
- All five teams total 9+15+17+19+21 = 81.
- After the first team starts, the rest split as boys + 3×boys = 4×boys, so the leftover must be a multiple of 4.
- Only removing 21 leaves 60, which splits as 15 boys and 45 girls (45 = 3×15), with whole teams 15 and 9+17+19.
- The started team has 21 members, choice (E).
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