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

Step-by-step solution

Let G. C. D. of (m, n)=d Then for some positive coprime integers x and y array ll & m=d x and n=d y & m+3 n-5=2 LCM (m, n)-11 GCD (m, n) & d x+3 d y-5=2 d x y-11 d or, & d(x+3 y-2 x y+11)=5 array Now, to maximize the sum m+n, d must be 5 aligned & x+3 y-2 x y+11=1 & or x+3 y-2 x y+10=0 & or (2 x-3)(2 y-1)=23 aligned aligned & Case I: 2 x-3=1 and 2 y-1=23 & x=2 and y=12 aligned This is not possible as x, y are coprime aligned & Case II : 2 x-3=23 and 2 y-1=1 & x=13 and y=1 & m+n=(13+1) 5=70 aligned