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Olympiad workbook
Exam
Olympiad workbook
Subject
IOQM
Chapter
Basics of Mathematics

Olympiad workbook · IOQM · Basics of Mathematics

Let m, n be natural numbers such that m+3 n-5=5 LCM(m, n)-11 GCD(m,...

Let m,nm, n be natural numbers such that m+3n−5=5LCM(m,n)−11GCD(m,n)m+3 n-5=5 \mathrm{LCM}(m, n)-11 \mathrm{GCD}(m, n) Find the maximum possible value of m+nm+n.

Correct answer

70

Step-by-step solution

Let G. C. D. of (m,n)=d(m, n)=d Then for some positive coprime integers xx and yy m=dx and n=dy∵m+3n−5=2 LCM (m,n)−11 GCD (m,n)∴dx+3dy−5=2dxy−11d or, d(x+3y−2xy+11)=5\begin{array}{ll} & m=d x \text { and } n=d y \\ \because & m+3 n-5=2 \text { LCM }(m, n)-11 \text { GCD }(m, n) \\ \therefore & d x+3 d y-5=2 d x y-11 d \\ \text { or, } & d(x+3 y-2 x y+11)=5 \end{array} Now, to maximize the sum m+n,dm+n, d must be 5 ∴x+3y−2xy+11=1 or x+3y−2xy+10=0 or (2x−3)(2y−1)=23\begin{aligned} & \therefore \quad x+3 y-2 x y+11=1 \\ & \text { or } \quad x+3 y-2 x y+10=0 \\ & \text { or } \quad(2 x-3)(2 y-1)=23 \end{aligned}  Case I:2x−3=1 and 2y−1=23∴x=2 and y=12\begin{aligned} & \text { Case } I: 2 x-3=1 \text { and } 2 y-1=23 \\ & \therefore \quad x=2 \text { and } y=12 \end{aligned} This is not possible as x,yx, y are coprime  Case II : 2x−3=23 and 2y−1=1∴x=13 and y=1∴m+n=(13+1)×5=70\begin{aligned} & \text { Case II : } 2 x-3=23 \text { and } 2 y-1=1 \\ & \therefore \quad x=13 \text { and } y=1 \\ & \therefore \quad m+n=(13+1) \times 5=70 \end{aligned}
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