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!

Question

Mathematics

Alec Berg

Today, 03:24

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!

+2

Answers (1)

Know the Answer?

Kaia Dorsey · Answer 1

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.