Try this beautiful problem from PRMO, 2019, Problem 18 based on Ordered Pairs.
How many ordered pairs \((a, b)\) of positive integers with \(a < b\) and \(100 \leq a\), \(b \leq 1000\) satisfy \(gcd (a, b) : lcm (a, b) = 1 : 495\) ?
Number theory
Orderd Pair
LCM
Answer:\(20\)
PRMO-2019, Problem 18
Pre College Mathematics
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 = \(13 + 7=20\)
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Why can't you consider p as 15 and q as 33
Sorry I got it.. Never mind