A pair of integers (m, n) satisfy m x n = gcd (m, n) + lcm (m, n) then m+n is:
Answers
Given : 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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