NUMBERS THAT ARE RELATIVELY PRIME (COPRIME) If two numbers have no common factor bigger than one, they are comparatively prime (coprime). The greatest common factor of relatively prime integers is 1, while the least common multiple is the product of these values. Prime numbers, on the whole, do not have **any common prime factors**. However, there are pairs of primes that have every other factor in their prime factorization, thus they have a unique factorization into factors which are not themselves products of **smaller pairs** of primes. For example, this is the case for 65537 and 895991.

NUMBERS THAT ARE NOT RELATIVELY PRIME A number is relatively prime to 0 if it has **no common factor** other than 1. Because 0 is itself a prime number, all positive integers are relatively prime to 0. Therefore, relative primeness is a property of numbers greater than or equal to 2.

A number is relatively prime to 1 if it has no common factor except 1. Because 1 is its own factorization, all positive integers are relatively prime to 1.

A number is not relatively prime to itself because if x is relatively prime to y, then either x or y must be even. If both x and y are odd, then there must be some integer z **such that** x + y = z.

When there are no common factors other than 1, two integers are relatively prime. This signifies that no other integer could equally divide **both values**. If gcd (a, b) = 1, two numbers a and b are said to be comparatively prime to each other. For example, the numbers 7 and 20 are relatively prime. There is no number that can be divided by 7 without losing **a whole number** of times, and 20 can be divided by 7 without leaving a remainder.

Relatively prime locations are important because they allow for flexibility in site selection. If you were to choose an absolute location for building sites, then it would be difficult if not impossible to change them later if conditions changed or new information became available. Relatively prime locations provide this flexibility because you can always move things around to see what works best. For example, if one site has good soil for growing vegetables but poor soil for **growing trees**, you could move the tree site over there and use the first site for vegetables. Or, if necessary, you could even move both sites somewhere else if needed.

There are many ways to find relatively prime locations. One simple method is to start with a location that is not prime itself and use its factors to create more prime locations. For example, if 0 is a factor of 10, 30, and 60, then those numbers are its factors. We can use these factors to create more 0's that are not equal to any other number so that they're guaranteed to be relatively prime.

If the largest common divisor of two positive integers is one, they are said to be relatively prime. For example, the numbers 10 and 7 are relatively prime since they share **no factors** other than 1. It's worth noting that neither integer has to be prime in order to be relatively prime. Both 8 and 15 are not primes, although they are comparatively primes.

Relatively prime numbers have many interesting properties. For example, there are an infinite number of pairs of relative primes, and every pair of positive integers can be expressed as a product of distinct relative primes. The proof for these statements can be found in any elementary number theory textbook.

It's also true that for **any two positive integers**, at least one of them is divisible by another. This follows immediately from the fact that if you divide both integers by the gcd, you will always get **a non-zero result**, which means at least one of them must be divided by it. For example, let's say that we want to prove that at least one of them is divisible by 4. We can do this by assuming that both are divisible by 4 and trying to find a contradiction. So, let's say that they are both divisible by 4. Then, 10%4 = 2 is even, so it cannot be a multiple of 4. This means that there must be at least one integer that is not divisible by 4, which contradicts our assumption that for all integers, either they are divisible by 4 or they are not.

Any two prime numbers can be checked to see if they are coprime. 2 and 3, for example, 5 and 7, 11 and 13, and so on. Co-prime numbers are all pairings of two consecutive numbers. Any two consecutive numbers have **a common factor** of one. The only exceptions are when these numbers are 1 and 2 or 2 and end with 1. These pairs of numbers are called coprime.

Coprimality is an important concept in number theory. It is possible to show that given any two positive integers, there are always at least one pair of coprime integers and one pair of non-coprime integers. For example, if we take the integers 4 and 9, there are no pairs of coprime integers because 4 and 9 have a common factor of 4. However, there are pairs of non-coprime integers: 4 and 9, and 7 and 21. Since 4, 7, 9, and 21 are all different numbers that are divisible by either 4 or 7, they cannot be both factors of any single integer. Therefore, they are pairs of non-coprime integers.

It is also possible to show that given any two positive integers, there are always at least one pair of coprime integers and one pair of non-coprime integers whose lengths are mutually exclusive, i.e., one length is not a multiple of the other.