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April 21, 2020

Smallest prime Problem | AIME I, 1999 | Question 1

Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1999 based on Smallest prime and arithmetic sequence.

Smallest prime Problem - AIME I, 1999


Find the smallest prime that is the fifth term of an increasing arithmetic sequence all four preceeding terms also being prime.

  • is 107
  • is 29
  • is 840
  • cannot be determined from the given information

Key Concepts


Smallest prime

Arithmetic Sequence

Algebra

Check the Answer


Answer: is 29.

AIME I, 1999, Question 1

Elementary Number Theory by David Burton

Try with Hints


First hint

Let the sequence be p, p+a, p+2a, p+3a, p+4a where p is prime and a is positive integer here p cannot be multiple of 2 or 3 or 4

Second Hint

then smallest p=5 taking a=6 we get a sequence of prime numbers

Final Step

5,11,17,23,29 then fifth term =29.

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