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 = $13 + 7=20$

Other useful links

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

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2 comments on “Ordered Pairs | PRMO-2019 | Problem 18”

Mukunda says:

August 31, 2023 at 7:41 pm

Why can't you consider p as 15 and q as 33

Reply
1. Mukunda says:
  
  August 31, 2023 at 7:42 pm
  
  Sorry I got it.. Never mind
  
  Reply

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