If gcd(a, b) = d And gcd(a, c) = d, Then gcd(a, b, c) = d

Let:

Then \(e \le d \), because there can be no integer that can divide \(a\) and \(b\) and also be greater than \(d\).

If \(\gcd(a, c) = d\), then \(d|a\), \(d|b\) and \(d|c\). Since \(e\) is the greatest common divisor of \(a\), \(b\) and \(c\), then \(e \ge d\).

If \(e \le d \) and \(e \ge d\), then \(e=d\).