Problem 27 · 2016 Math Kangaroo
Stretch
Number Theory
primes
Jakob writes down four consecutive positive whole numbers. He calculates all possible sums of three of those numbers and realises that none of those sums is a prime number. What is the smallest number that Jakob could have written down?
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Answer: C — 7
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Hint 1 of 2
Adding three of four consecutive numbers leaves out one; write the four sums.
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Hint 2 of 2
Two of the sums are automatically multiples of 3; make the other two composite too.
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Approach: search the smallest start
- For a start n the four sums are 3n+3, 3n+4, 3n+5, 3n+6; the first and last are multiples of 3.
- Need 3n+4 and 3n+5 composite as well; the first n that works is n = 7 (25 and 26).
- So the smallest number written is 7.
Mark:
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