Question

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?
I would greatly appreciate it if someone responded!

Answer 1

No, It is not necessarily true that the greatest common divisor of all three numbers is greater than 1.

Explanation:

Consider the provided information.

It is given that Aisha's number and Benoit's number have a common divisor greater than 1.

Let us assume any number having common divisor greater than 1

Let say Aisha's number is 10 and Benoit's number is 14.

Now the common divisor in both the numbers are:

Aisha: 10 = 2×5

Benoit: 14 = 2×7

Here, the common divisor is 2.

Now, it is given that Aisha's number and Carleen's number also have a common divisor greater than 1.

Let us assume Carleen's number is 35. Thus the common divisor in both the numbers are:

Aisha: 10 = 2×5

Carleen: 35 = 5×7

Here, the common divisor is 5.

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

Benoit: 14 = 2×7

Carleen: 35 = 5×7

Here, the common divisor is 7.

Now, we need to find that the greatest common divisor of all three numbers is greater than 1.

Aisha: 10 = 2×5

Benoit: 14 = 2×7

Carleen: 35 = 5×7

There is no common divisor greater than 1.

Hence, it is not necessarily true that the greatest common divisor of all three numbers is greater than 1.

Answer 2

Question:
If
GCD(A,B)>1
GCD(B,C)>1
GCD(C,A)>1
is it *always* true that GCD(A,B,C)>1 ?

Answer: no
Following is a counter example: A=6=2*3, B=14=2*7, C=21=3*7
There is no GCD greater than 1 that divides ALL of A, B and C.
I am sure you will be able to find other counter examples.