Problem 30 · 2009 Math Kangaroo
Stretch
Number Theory
primescareful-counting
A single-digit prime number is called “strange.” A prime number with more than one digit is called “strange” if the numbers obtained by deleting its first digit and by deleting its last digit are both strange prime numbers again. How many strange prime numbers are there?
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Answer: D — 9
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Hint 1 of 2
Start from the single-digit 'strange' primes 2, 3, 5, 7 and build longer ones digit by digit.
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Hint 2 of 2
A longer prime is strange only if dropping its first digit AND dropping its last digit each give a strange prime.
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Approach: build strange primes upward by length
- The single-digit strange primes are 2, 3, 5, 7.
- A two-digit prime is strange if removing either end digit leaves a strange prime: 23, 37, 53, 73 qualify.
- A three-digit prime needs both trimmed numbers strange: only 373 works (37 and 73 are strange).
- No four-digit prime qualifies, so the count is 4 + 4 + 1 = 9.
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