Question

A pair of integers (m, n) satisfy mxn= gcd (m, n)+lem (m, n) then m+n is:​

Answer 1

Give : A pair of integers (m, n) satisfy m x n= gcd (m, n)+ lcm (m, n)  

To Find :   m+n

Solution:

m x n= gcd (m, n)+ lcm (m, n)  

also we know that

m x n= gcd (m, n) *  lcm (m, n)  

gcd (m, n) + lcm (m, n)   = gcd (m, n)  *  lcm (m, n)

where lcm (m, n) = k .  gcd (m, n)

=> gcd (m, n) +  k .  gcd (m, n)   = gcd (m, n)  *  k .  gcd (m, n)

=> gcd (m, n) ( k + 1)  = gcd (m, n)  k .  gcd (m, n)

=>  ( k + 1)  =  gcd (m, n)  k

=>   1 +  1/k  = gcd (m, n)

as gcd (m, n) is integer hence k  must be 1  

so   lcm (m, n) =      gcd (m, n)

this is only possible when   m =  n

=>    lcm (m, n) =      gcd (m, n)   = m

=> m²  =  m  +  m

=> m² = 2m

=>   m  = 2

Hence pair of integers   ( 2 , 2)

where

m x n= gcd (m, n)+ lcm (m, n)  

2 * 2  =  2  +  2

=> 4 = 4

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