Problem:
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.

Question

Problem:
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.

SmokeyBrown · Answer

So, we have three positive numbers, A, B, and C.
We are told in the problem that each of the three numbers share at least one common divisor (greater than 1) with each of the other three numbers. 
Now, the problem asks us if the three numbers must have a Greatest Common Divisor greater than one. To prove or disprove this, we can look at the opposite case. What would have to be true if the three numbers did not have a Greatest Common Divisor greater than 1? This would mean that at least one of the number pairs do not share a common divisor greater than 1.
As we know, all three number pairs do share a common divisor greater than 1, so the case where all three numbers do not share a greater-than-one common divisor cannot exist.

Hero · Answer

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Hero · Answer

In this case, let a = 6, b = 10, and c = 15. 
The common divisor of a and b is 2.
The common divisor of a and c is 3.
The common divisor of b and c is 5.
But the greatest common divisor of a, b and c is not greater than 1.

Hero · Answer

@SmokeyBrown

Vocaloid · Answer

@Hero this is an interesting problem, may I ask how you came up with those 3 numbers? is there a systematic way to do it?

Hero · Answer

What I did was I figured that if there was a way to disprove it, it would have to involve prime numbers so I decided to use prime numbers as the factors of a, b, and c and then go from there.

Vocaloid · Answer

ah, now I see it, that's pretty smart, well done

SmokeyBrown · Answer

Ooh, very nice.