Final answer:
The question seems to be about gcd properties where we prove that m and the gcd of m and n are coprime to n. This is concluded by the logic that any integer dividing both n and gcd(m, n) must divide the gcd itself, and therefore must be 1.
Step-by-step explanation:
The question appears to be about the properties of the greatest common divisor (gcd). However, the question seems to have a typo and might be asking if the gcd of m and gcd of m and n (which is the same as gcd of m and n) is coprime to n. Assuming this interpretation is correct, the proof is based on the fundamental properties of gcd.
To prove that gcd(m, gcd(m, n)), n gcd(m, n) = 1, we start by noting that gcd(m, n) divides both m and n, hence gcd(m, gcd(m, n)) is equal to gcd(m, n). Since n and gcd(m, n) are divisors of each other, they only share the factors they have in common. Therefore, if any integer divides both n and gcd(m, n), it must divide the gcd by definition, implying that the gcd is the largest such integer.
To state it more formally, if d is a common divisor of n and gcd(m, n), then d divides the gcd(m, n), i.e., d divides itself, which implies that d must equal 1, since any common divisor cannot be greater than gcd(m, n). Hence, gcd(m, gcd(m, n)), n gcd(m, n) must indeed equal 1, showing that they are coprime.