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
But try the problem first...
Answer:\(20\)
PRMO-2019, Problem 18
Pre College Mathematics
First hint
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 ..........
Second Hint
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........
Final Step
Therefore Total solutions = \(13 + 7=20\)
AMC - AIME Boot Camp... for brilliant students.
Use our exclusive one-on-one plus group class system to prepare for Math Olympiad
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
But try the problem first...
Answer:\(20\)
PRMO-2019, Problem 18
Pre College Mathematics
First hint
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 ..........
Second Hint
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........
Final Step
Therefore Total solutions = \(13 + 7=20\)
AMC - AIME Boot Camp... for brilliant students.
Use our exclusive one-on-one plus group class system to prepare for Math Olympiad
