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........
