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Algebra Arithmetic Math Olympiad PRMO

Smallest Positive Integer | PRMO 2019 | Question 14

Try this beautiful problem from the Pre-RMO, 2019 based on Smallest Positive Integer. You may use sequential hints to solve the problem.

Try this beautiful problem from the PRMO, 2019 based on Smallest Positive Integer.

Smallest Positive Integer – PRMO 2019


Find the smallest positive integer n\(\geq\)10 such that n+6 is a prime and 9n+7 is a perfect square.

  • is 107
  • is 53
  • is 840
  • cannot be determined from the given information

Key Concepts


Integers

Primes

Perfect Square

Check the Answer


Answer: is 53.

PRMO, 2019, Question 14

Elementary Number Theory by David Burton

Try with Hints


First hint

Let 9n+7=\(m^{2}\) n+6 prime then n+6 odd then n is odd then n=2k+1 then 9(2k+1)+7=\(m^{2}\) then 18k=\(m^{2}\)-16=(m+4)(m-4) then 18k even m is even then m=2p

Second Hint

18k=(2p+4)(2p-4)=4(p+2)(p-2) then 9k=2(p+2)(p-2)then k even then k=2d then 18d=2(p+2)(p-2) then 9d=(p+2)(p-2) then p of form 9q+2,9q-2

Final Step

for p=9q-2 then m=2(9q-2) for q=1 then\(m^{2}\)=196then n=21 then n+6=27 non prime, for p=9q+2 then m=2(9q+2) for q=1 \(m^{2}\)=484 then n=53 then n+6=59 prime then n=53.

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