Question

A list consists of the following pairs of numbers: \[51,53;55,57;59,61;63,65;67,69;71,73\]. Categorize them as pairs of co-primes.

Answer 1

Hint: A pair of numbers is said to be coprime if their highest common factor is one. We can check the common factors of the given factors. If there is any other common factor other than one, they will not be coprime and otherwise they will be co-prime.

Complete step-by-step answer:
Given the pairs of numbers \[51,53;55,57;59,61;63,65;67,69;71,73\]
We have to categorize them as co-primes.
A pair of numbers is said to be coprime if their highest common factor is one.
Let us check each pair one by one.
1) $51,53$
Here one is prime and the other is composite.
We can see that these two numbers have no common factor other than one.
This gives these numbers are co-primes.
2) $55,57$
Here both are composite numbers.
But these two numbers have no common factor other than one.
This gives these numbers are co-primes.
3) $59,61$
Here both are prime numbers.
We can see that these two numbers have no common factor other than one.
This gives these numbers are co-primes.
4) $63,65$
Here both are composite numbers.
We can see that these two numbers have no common factor other than one.
This gives these numbers are co-primes.
5) $67,69$
Here one is prime and the other is composite.
We can see that these two numbers have no common factor other than one.
This gives these numbers are co-primes.
6) $71,73$
Here both are prime numbers.
We can see that these two numbers have no common factor other than one.
This gives these numbers are co-primes.
So we can see that all the given pairs are co-primes.

Note: Here we checked whether the pairs are co-primes. If both or at least one is a prime, then that pair will be clearly co-prime. If both the numbers are composite, they may have other common factors as well.