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# Smallest Positive Integer | PRMO 2019 | Question 14

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

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.

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