Problem
NT-B1-M02-P024 Sum and LCM
#24
★★★★★ Level 5 of 5
Find two positive integers \(a,b\) if \(a+b=154\) and \(\operatorname{lcm}(a,b)=840\).
Method inspiration: local number theory source
First prove that \(\gcd(a,b)=\gcd(a+b,\operatorname{lcm}(a,b))\).
Let \(d=\gcd(a,b)\), \(a=dx\), \(b=dy\), where \(\gcd(x,y)=1\). Then \(\operatorname{lcm}(a,b)=dxy\), and \(a+b=d(x+y)\). Since \(\gcd(x+y,xy)=1\), we get \(\gcd(a+b,\operatorname{lcm}(a,b))=d\). Therefore \(d=\gcd(154,840)=14\).
Then \(a=14x\), \(b=14y\), \(x+y=11\), and \(\operatorname{lcm}(a,b)=14xy=840\), so \(xy=60\). The numbers \(x,y\) are roots of \(t^2-11t+60=0\), giving \(\{x,y\}=\{5,6\}\). Hence \(\{a,b\}=\{70,84\}\).
Level 5 for Book 1: a hidden lemma about the GCD of the sum and LCM, followed by reconstruction.