Bunuel
If a, b, and c are positive integers, what is the greatest common divisor of a, b and c?
(1) The greatest common divisor of a and b is 3
(2) The greatest common divisor of b and c is 4
Official Solution:If \(a\), \(b\), and \(c\) are positive integers, what is the greatest common divisor of \(a\), \(b\) and \(c\)? (1) The greatest common divisor of \(a\) and \(b\) is 3
If the third unknown, \(c\), is also divisible by 3, then the greatest common divisor of \(a\), \(b\) and \(c\) will be 3 (it cannot be more than 3, since 3 is the GCD of \(a\) and \(b\)) but if \(c\) is NOT divisible by 3, then the greatest common divisor of \(a\), \(b\) and \(c\) will be 1. Not sufficient.
(2) The greatest common divisor of \(b\) and \(c\) is 4
If the third unknown, \(a\), is also divisible by 4, then the greatest common divisor of \(a\), \(b\) and \(c\) will be 4 (it cannot be more than 4, since 4 is the GCD of \(b\) and \(c\)) but if \(a\) is NOT divisible by 4, then the greatest common divisor of \(a\), \(b\) and \(c\) will be 1. Not sufficient.
(1)+(2) From (1) we know that \(b\) is divisible by 3 and if \(c\) were also divisible by 3, then the greatest common divisor of \(b\) and \(c\) would be 12, not 4, as given in (2). Thus, \(c\) is NOT divisible by 3, and as we deduced in (1), if \(c\) is NOT divisible by 3, then the greatest common divisor of \(a\), \(b\) and \(c\) will be 1. Sufficient.
Answer: C