How Cheenta works to ensure student success?
Explore the Back-Story

Smallest Positive Integer | PRMO 2019 | Question 14

Join Trial or Access Free Resources

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


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

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

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.

Subscribe to Cheenta at Youtube


Leave a Reply

Your email address will not be published. Required fields are marked *

This site uses Akismet to reduce spam. Learn how your comment data is processed.

Knowledge Partner

Cheenta is a knowledge partner of Aditya Birla Education Academy
Cheenta

Cheenta Academy

Aditya Birla Education Academy

Aditya Birla Education Academy

Cheenta. Passion for Mathematics

Advanced Mathematical Science. Taught by olympians, researchers and true masters of the subject.
JOIN TRIAL
support@cheenta.com
Trial
Math Olympiad Program
magic-wandrockethighlight