Give an example of four integers which are relatively prime but not relatively prime in pairs
I can think of 3 integers that are relatively prime but not relatively prime in pairs: 10,12,15.
(10,12,15)=1
(10,12)=2
(12,15)=3
(10,15)=5
But I am not able to think of an example for 4 integers that are relatively prime but no relatively prime in pairs…Can somebody help me with this.
Best Answer
For the three number case, many different examples can be found by the following construction: let $p,q,r$ be distinct primes. Then $pq, qr, pr$ will have the property you want.
Here's how you might come up with an answer. Let $p,q,r,s$ be distinct primes. What can you say about the numbers $pqr, pqs, prs, qrs$? Well, none of $p,q,r,s$ divide all four of them, however its not too hard to see that any two share a common factor.
Explicitly, choosing the first four primes you can see that $30, 42, 70, 105$ should work.