Problem 16 · 2018 Math Kangaroo
Medium
Number Theory
primescasework
Alice writes down three prime numbers that are all less than 100. She only uses the digits 1, 2, 3, 4 and 5, and she uses each digit exactly once. Which of the following prime numbers did she definitely write down?
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Answer: D — 41
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Hint 1 of 2
Five digits split as a 1-digit prime and two 2-digit primes; even digits can't end a prime.
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Hint 2 of 2
Place 2 and 4 as tens-digits, then find the forced prime common to every valid arrangement.
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Approach: place the even digits and find what is forced
- Using digits 1,2,3,4,5 once, the even digits 2 and 4 can only be tens-digits (a prime can't end in an even digit).
- Working through the options gives sets like {5, 23, 41} and {2, 41, 53}.
- The prime that appears in every valid set is 41, so she definitely wrote it.
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