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# PRMO 2016 Problem 1 | Number Theory Problem

Try this Number Theory Problem based on finding the number of solutions from PRMO - 2016.

## Number Theory Problem - PRMO 2016 Problem 1

Consider all possible integers $n \geq 0$ such that
$$\left(5 \times 3^{m}\right)+4=n^{2}$$
holds for some corresponding integer $m \geq 0$. Find the sum of all such $n$.

### Key Concepts

GCD of Numbers

Power of Primes

Parity of a Number

## Suggested Book | Source | Answer

Elementary Number Theory by David Burton

Excursion in mathematics

PRMO 2016

The required answer is 10

## Try with Hints

$\left(5 \times 3^{m}\right)+4=n^{2}$

$\rightarrow \left(5 \times 3^{m}\right) = n^{2}-4$

$\rightarrow \left(5 \times 3^{m}\right) = (n-2)(n+2)$

Observe that $gcd (n-2,n+2)$ = $1$ or $2$ or $4$

But $\left(5 \times 3^{m}\right) = (n-2)(n+2)$

Hence both $(n-2)$ , $(n+2)$ are odd.

Hence one possible case:

$(n-2)$ = $5$

$(n+2)$ = $3^{m}$

$\rightarrow$ $4$ = $3^{m}$ - $5$

$\rightarrow$ $m$ = $2$

$\rightarrow$ $n$ = $7$

Similarly find other values of n and add

Math Olympiad Program at Cheenta

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Try this Number Theory Problem based on finding the number of solutions from PRMO - 2016.

## Number Theory Problem - PRMO 2016 Problem 1

Consider all possible integers $n \geq 0$ such that
$$\left(5 \times 3^{m}\right)+4=n^{2}$$
holds for some corresponding integer $m \geq 0$. Find the sum of all such $n$.

### Key Concepts

GCD of Numbers

Power of Primes

Parity of a Number

## Suggested Book | Source | Answer

Elementary Number Theory by David Burton

Excursion in mathematics

PRMO 2016

The required answer is 10

## Try with Hints

$\left(5 \times 3^{m}\right)+4=n^{2}$

$\rightarrow \left(5 \times 3^{m}\right) = n^{2}-4$

$\rightarrow \left(5 \times 3^{m}\right) = (n-2)(n+2)$

Observe that $gcd (n-2,n+2)$ = $1$ or $2$ or $4$

But $\left(5 \times 3^{m}\right) = (n-2)(n+2)$

Hence both $(n-2)$ , $(n+2)$ are odd.

Hence one possible case:

$(n-2)$ = $5$

$(n+2)$ = $3^{m}$

$\rightarrow$ $4$ = $3^{m}$ - $5$

$\rightarrow$ $m$ = $2$

$\rightarrow$ $n$ = $7$

Similarly find other values of n and add

Math Olympiad Program at Cheenta

## Subscribe to Cheenta at Youtube

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Cheenta is a knowledge partner of Aditya Birla Education Academy