2022/080) Let m and n be positive integers such that gcd(m,n) + lcm (m,n) = m + n. Show that one of the 2 numbers is divisible by the other

Let gcd(m,n) = p.

then m = pq and n = pr for some q and r and gcd(q,r) = 1

gcd(m,n)  = p as we have chosen

lcm(m, n) = pqr as q and r are co-primes

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

$=>p + pqr = pq + pr$

$=>1 + qr = q + r$

$=>qr - q -r + 1= 0$

$=>(q-1)(r-1) = 0$

q =1 mean n is divisible by m 

or r =1 meand m is divisible byn

hence proved