 Any number that divides "a" and "b" must also divide "c" so every common divisor of "a" and "b" is also a common divisor of "b" and "c".Similarly, "a" = "b" + "c" and every common divisor of "b" and "c" is also a common divisor of "a" and "b".So the two pairs ("a", "b") and ("b", "c") have the same common divisors, and thus gcd("a","b") = gcd("b","c").