Question

Look at the following pairs of numbers and determine whether they are co-prime or prime:
a) 8,9
b) 11,13
c) 15,16
d) 64,65
e) 71,73

Answer 1

Hint: A prime number is defined as a number that has no factor other than 1 and itself. While two integers a and b are said to be coprime if the only positive integer that divides both of them is 1.
So, we need to find the factors of the given pairs by factorization method and compare them with the given definition of prime numbers and co-prime numbers.

Complete step by step answer:
a) 8,9
Factors of 8 are: $1\times 2\times 2\times 2$
Factors of 9 are: $1\times 3\times 3$
Since the pair is commonly divisible by 1, so they are co-prime numbers.
But both the numbers have factors including 1 and itself, so they are not prime numbers.

b) 11,13
Factors of 11 are: $1\times 11$
Factors of 13 are: $1\times 13$
Since, both the numbers do not have factors except 1 and itself, so they are not prime numbers.
Also, the pair is commonly divisible by 1, so they are co-prime numbers.
c) 15,16
Factors of 15 are: $1\times 3\times 5$
Factors of 16 are: $1\times 2\times 2\times 2\times 2$
Since, both the numbers have factors including 1 and itself, so they are not prime numbers.
Also, the pair is commonly divisible by 1, so they are co-prime numbers.

d) 64,65
Factors of 64 are: $1\times 2\times 2\times 2\times 2\times 2\times 2$
Factors of 65 are: \[1\times 3\times 5\times 13\]
Since, both the numbers have factors including 1 and itself, so they are not prime numbers.
Also, the pair is commonly divisible by 1, so they are co-prime numbers.

d) 71,73
Factors of 71 are: $1\times 71$
Factors of 73 are: \[1\times 73\]
Since, both the numbers do not have factors except 1 and itself, so they are prime numbers.
Also, the pair is commonly divisible by 1, so they are co-prime numbers.

Note: Do not assume that every prime number is a co-prime number as their definition is a bit confusing. So, to identify correctly between prime numbers and co-prime numbers always find the factors of the numbers and then compare the factors.