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Determine-x-amp-y-such-that-lcm-x-y-gcd-x-y-x-y-

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Question Number 106017 by Rasheed.Sindhi last updated on 02/Aug/20
Determine x & y, such that: lcm(x,y)−gcd(x,y)=x+y.
Determinex&y,suchthat:lcm(x,y)−gcd(x,y)=x+y.
Answered by Rasheed.Sindhi last updated on 03/Aug/20
Let gcd(x,y)=k and x=pk , y=qk with gcd(p,q)=1 i-e p,q are coprime. lcm(x,y)−gcd(x,y)=x+y lcm(pk,qk)−gcd(pk,qk)=pk+qk ⇒pqk−k=pk+qk ⇒pq−1=p+q ⇒p=2 , q=3 ∨ p=3 , q=2 ⇒x=2k ,y=3k ∨ x=3k,y=2k {x,y}={2k,3k}
Letgcd(x,y)=kandx=pk,y=qkwithgcd(p,q)=1i−ep,qarecoprime.lcm(x,y)−gcd(x,y)=x+ylcm(pk,qk)−gcd(pk,qk)=pk+qk⇒pqk−k=pk+qk⇒pq−1=p+q⇒p=2,q=3∨p=3,q=2⇒x=2k,y=3k∨x=3k,y=2k{x,y}={2k,3k}

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