Aisha, Benoit, and Carleen are each thinking of a positive integer.

Aisha's number and Benoit's number have a common divisor greater than 1.
Aisha's number and Carleen's number also have a common divisor greater than 1.
Benoit's number and Carleen's number also have a common divisor greater than 1.

Is it necessarily true that the greatest common divisor of all three numbers is greater than 1?

If it is necessarily true, explain how you know it's true.

If it isn't necessarily true, convince us by showing us three numbers for which it's false -- in other words, show us three integers such that every pair have a common divisor greater than 1, but all three numbers don't have a common divisor greater than 1.