Question

Which of the following numbers are co-prime:
A. $ 18 $ and $ 35 $
B. $ 15 $ and $ 37 $
C. $ 30 $ and $ 415 $
D. $ 17 $ and $ 168 $
E. $ 216 $ and $ 215 $
F. $ 81 $ and $ 16 $

Answer 1

Hint: To solve this question we should first know what are co-prime numbers. A co-prime number is a set of numbers or integers which have only one factor as their common factor i.e. $ 1 $ , it means that their highest common factor or HCF will be $ 1 $ . Here we will check each of the pairs and their common factors and then solve them.

Complete step-by-step answer:
As we have the pairs of numbers, we will solve them one by one.

A.Here we have $ 18 $ and $ 35 $ . We will write their factors.
So the factor of $ 18 = 1,2,3,6,9,18 $ and factor of $ 35 = 1,5,7,35 $ . We can see that both have only one common factor. Therefore they are co-prime numbers.

B.Here we have $ 15 $ and $ 37 $ . We will write their factors.
So the factor of $ 15 = 1,3,5,15 $ and factor of $ 37 = 1,37 $ . We can see that both have only one common factor. Therefore they are co-prime numbers.

C.Here we have $ 30 $ and $ 415 $ . We will write their factors.
So the factor of $ 30 = 1,2,3,5,6,15,30 $ and factor of $ 35 = 1,5,7,35 $ . We can see that both have more than one common factor i.e. $ 1,5 $ . Therefore they are not co-prime numbers.

D.Here we have $ 17 $ and $ 68 $ . We will write their factors.
So the factor of $ 17 = 1,17 $ and factor of $ 68 = 1,2,4,17,34,68 $ . In this also we can see that both have more than one common factor. Therefore they are not co-prime numbers.

E.Here we have $ 216 $ and $ 215 $ . We will write their factors.
So the factor of $ 216 = 1,2,3,4,6,8,9,12,18,24,27,36,54,72,102,216 $ and factor of $ 215 = 1,5,43,215 $ . We can see that both have only one common factor. Therefore they are co-prime numbers.

F.Here we have $ 81 $ and $ 16 $ . We will write their factors.
So the factor of $ 81 = 1,3,9,27,81 $ and factor of $ 16 = 1,2,4,8,16 $ . We can see that both have only one common factor. Therefore they are co-prime numbers.


Note: We should note that prime numbers and co-prime numbers, both are different. A prime number is defined as a number which has no factor other than $ 1 $ and itself. But the co-prime numbers are considered in pairs and two numbers are co-prime if they have a common factor i.e. $ 1 $ .