Problem 7 · 2025 Math Kangaroo
Easy
Number Theory
primescasework
Sarah has a bag of 18 balls numbered from 1 to 18. What is the smallest number of balls Sarah must remove from the bag to be sure that she has removed at least three prime numbers?
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Answer: D — 14
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Hint 1 of 2
Ask how many balls are NOT prime — those could all come out before any prime.
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Hint 2 of 2
After removing every non-prime, the next few must be primes; worst case removes all non-primes first.
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Approach: worst-case (pigeonhole) counting
- Primes from 1–18: 2,3,5,7,11,13,17 — seven of them; the other 11 numbers are non-prime.
- Worst case she draws all 11 non-primes first, then needs 3 more to guarantee three primes.
- Smallest guaranteed count = 11 + 3 = 14.
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