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Ordered Pairs | PRMO-2019 | Problem 18

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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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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