Question

Find a set of four mutually relatively prime integers such that no two of them are relatively prime.

Answer 1

To find four mutually relatively prime integers with no two relatively prime, you can multiply a fixed prime number by different distinct primes. Example set provided: {2, 6, 10, 14}.

Step-by-step explanation:

The question involves finding a set of four mutually relatively prime integers where no two are relatively prime to each other. This means each of the four integers must share a common prime factor with each of the others. Here's how you might approach finding such a set:

1. Pick a prime number, let's say 2.
2. Multiply this prime number by different distinct prime numbers to create the set. For example, you could use 2, 2×3, 2×5, and 2×7.
3. Verify that each pair of numbers in the set shares the common factor of 2, confirming that they are not relatively prime to each other.
4. However, when you consider the set as a whole, the numbers are mutually relatively prime because no number outside the set has a common factor with all of them except 1.

Using this method, an example set would be {2, 6, 10, 14}. Each pair of numbers within this set shares the common factor 2, so none of the pairs are relatively prime. However, there is no number greater than 1 that divides all four numbers, making the set mutually relatively prime.