Try this beautiful problem from PRMO, 2019, Problem 18 based on Ordered Pairs.

## Orderd Pairs | PRMO | Problem-18

How many ordered pairs \((a, b)\) of positive integers with \(a < b\) and \(100 \leq a\), \(b \leq 1000\) satisfy \(gcd (a, b) : lcm (a, b) = 1 : 495\) ?

- $20$
- $91$
- $13$
- \(23\)

**Key Concepts**

Number theory

Orderd Pair

LCM

## Check the Answer

But try the problem first…

Answer:\(20\)

PRMO-2019, Problem 18

Pre College Mathematics

## Try with Hints

First hint

At first we assume that \( a = xp\)

\(b = xq\)

where \(p\) & \(q\) are co-prime

Therefore ,

\(\frac{gcd(a,b)}{LCM(a ,b)} =\frac{495}{1}\)

\(\Rightarrow pq=495\)

Can you now finish the problem ……….

Second Hint

Therefore we can say that

\(pq = 5 \times 9 \times 11\)

\(p < q\)

when \( 5 < 99\) (for \(x = 20, a = 100, b = 1980 > 100\)),No solution

when \(9 < 55\) \((x = 12\) to \(x = 18)\),7 solution

when,\(11 < 45\) \((x = 10\) to \(x = 22)\),13 solution

Can you finish the problem……..

Final Step

Therefore Total solutions = \(13 + 7=20\)

## Other useful links

- https://www.cheenta.com/largest-possible-value-prmo-2019-problem-17/
- https://www.youtube.com/watch?v=fRj9NuPGrLU&t=282s

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