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
Answer:\(20\)
PRMO-2019, Problem 18
Pre College Mathematics
Try with Hints
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 ……….
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……..
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