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.
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.

  • 2
    $\begingroup$ $(10,6,15, 2\cdot3\cdot5\cdot7)$? The trick is to write up the prime factorization of the numbers and make sure that any two has at least one common factor, while all the four don't have a common factor. $\endgroup$ – Andrew Sep 28 '16 at 12:01

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.


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