Question

Show that the following pairs are co-prime $59,97$.

Answer 1

Hint: Co-prime number are the number which have highest common factors ( HCF ) equals to $1$ so for this as the factor of $59 = 1 \times 59$ because $59$ is prime number and $97 = 1 \times 97$ because $97$ is prime number.

Complete step-by-step answer:
A Co-prime number is a set of the numbers or integers which have only $1$ as their common factor that is their highest common factor (HCF) will be $1$ .
Means that if both the numbers is divisible by $1$ only not any other number can divide it without leaving reminder ,
So we have to show that the following pairs are co-prime $59,97$
By prime factorization method ,
$59$ can be written as $59 = 1 \times 59$ because $59$ is prime number .
now
$97$ can be written as $97 = 1 \times 97$ because $97$ is prime number .
So from the prime factorization method only $1$ is the common factor or HCF of both numbers is $1$

Hence the given pair 59 and 97 are co-prime numbers.

Note: As from above the pairs of prime numbers are always coprime but it is not necessary that a pair of co-prime numbers is prime number for example $7,15$ .
so $7 = 1 \times 7$ it is a prime number and for $15 = 1 \times 3 \times 5$ so only $1$ is the common factor or HCF of both numbers is $1$
Hence it is a co-prime number but both numbers are not prime .
$1$ is co-prime with every number.