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Ordered Pairs | PRMO-2019 | Problem 18

Try this beautiful Number Theory problem from PRMO, 2019, problem-18, based on Ordered Pairs. You may use sequential hints to solve the problem.

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 = \(13 + 7=20\)

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