It is called relatively prime (coprime or are relatively prime to each other) to any pair of integers have no common divisor other than 1.
In other words, two integers are relative primes if in their decompositions into prime numbers, they do not have any factor in common.
For example, if 4 and 25 are chosen, the prime factorizations of each are 2² and 5² respectively. As can be seen, these do not have any common factors, therefore 4 and 25 are relative primes.
On the other hand, if 6 and 24 are chosen, when performing their decompositions into prime factors, we obtain that 6 = 2 * 3 and 24 = 2³ * 3.
As you can see, these last two expressions have at least one factor in common, therefore, they are not relative primes.
Relative Cousins
One detail to be careful with is that saying that a pair of integers are relative primes does not imply that any of them is a prime number.
On the other hand, the definition above can be summarized as follows: two integers "a" and "b" are relative primes if, and only if, the greatest common divisor of these is 1, that is, gcd (a, b) = 1.
Two immediate conclusions from this definition are that:
-If «a» (or «b») is a prime number, then gcd (a, b) = 1.
-If «a» and «b» are prime numbers, then gcd (a, b) = 1.
That is, if at least one of the chosen numbers is a prime number, then directly the pair of numbers are relative primes.
Other features
Other results that are used to determine if two numbers are relative primes are:
-If two integers are consecutive then they are relative primes.
-Two natural numbers "a" and "b" are relative primes if, and only if, the numbers "(2 ^ a) -1" and "(2 ^ b) -1" are relative primes.
-Two integers «a» and «b» are relative primes if, and only if, when graphing the point (a, b) in the Cartesian plane, and constructing the line that passes through the origin (0,0) and (a, b), it does not contain any point with integer coordinates.
Examples
1.- Consider the integers 5 and 12. The decompositions in prime factors of both numbers are: 5 and 2² * 3 respectively. In conclusion, gcd (5,12) = 1, therefore, 5 and 12 are relative primes.
2.- Let the numbers -4 and 6. Then -4 = -2² and 6 = 2 * 3, so that the LCD (-4,6) = 2 ≠ 1. In conclusion -4 and 6 are not relative primes.
If we proceed to graph the line that passes through the ordered pairs (-4.6) and (0,0), and to determine the equation of said line, it can be verified that it passes through the point (-2,3).
Again it is concluded that -4 and 6 are not relative primes.
3.- The numbers 7 and 44 are relative primes and it can be concluded quickly thanks to the above, since 7 is a prime number.
4.- Consider the numbers 345 and 346. Being two consecutive numbers it is verified that gcd (345,346) = 1, therefore 345 and 346 are relative primes.
5.- If the numbers 147 and 74 are considered, then these are relative primes, since 147 = 3 * 7² and 74 = 2 * 37, therefore the LCD (147,74) = 1.
6.- The numbers 4 and 9 are relative primes. To demonstrate this, the second characterization mentioned above can be used. Indeed, 2 ^ 4 -1 = 16-1 = 15 and 2 ^ 9-1 = 512-1 = 511.
The numbers obtained are 15 and 511. The prime factorizations of these numbers are 3 * 5 and 7 * 73 respectively, so that gcd (15,511) = 1.
As you can see, using the second characterization is a longer and more laborious job than verifying it directly.
7.- Consider the numbers -22 and -27. Then these numbers can be rewritten as follows: -22 = -2 * 11 and -27 = -3³. Therefore, the gcd (-22, -27) = 1, so -22 and -27 are relative primes.
References
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