Question

Which of the following pairs of numbers are relatively prime?
A) 36 and 54
B) 52 and 78
C) 54 and 114
D) 59 and 61

Answer 1

Hint: We will first find factors of all the numbers option by option. Then after finding the factors, we will find the HCF of numbers in pairs. If the HCF of the numbers in any of the options comes to be 1, we have the required answer.

Complete step-by-step answer:
We have four options with us. Let us go through them, one by one:-
Option A : 36 and 54
We see that:
$ \Rightarrow 36 = 2 \times 2 \times 3 \times 3$
$ \Rightarrow 54 = 2 \times 3 \times 3 \times 3$
We can clearly observe that, if we start making out pairs of common multiples from the factors of both the numbers, we will get 2, 3 and 3.
Therefore, \[HCF\left( {36,54} \right) = 2 \times 3 \times 3 = 18\].
Since their HCF is 18, they are not relatively prime.
Option B : 52 and 78
We see that:
$ \Rightarrow 52 = 2 \times 2 \times 13$
$ \Rightarrow 78 = 2 \times 3 \times 13$
We can clearly observe that, if we start making out pairs of common multiples from the factors of both the numbers, we will get 2 and 13.
Therefore, \[HCF\left( {52,78} \right) = 2 \times 13 = 26\].
Since their HCF is 26, they are not relatively prime.
Option C : 54 and 114
We see that:
$ \Rightarrow 54 = 2 \times 3 \times 3 \times 3$
$ \Rightarrow 114 = 2 \times 3 \times 19$
We can clearly observe that, if we start making out pairs of common multiples from the factors of both the numbers, we will get 2 and 3.
Therefore, \[HCF\left( {54,114} \right) = 2 \times 3 = 6\].
Since their HCF is 6, they are not relatively prime.
Option D : 59 and 61
We see that:
$ \Rightarrow 59 = 1 \times 59$
$ \Rightarrow 61 = 1 \times 61$
We can clearly observe that, if we start making out pairs of common multiples from the factors of both the numbers, we will get 1.
Therefore, \[HCF\left( {59,61} \right) = 1\].
Since, their HCF is 1, they are relatively prime.

Hence, the correct answer is (D) 59 and 61.

Note: The students must wonder why we find HCF of numbers and say that they are relatively prime. HCF stands for highest common factor. If two factors have the highest common factor as 1, that means no other number than 1 can divide both of them. If only 1 is the common divisor of two numbers, then they are known to be coprime or relatively prime.
The students must also know that relatively prime does not imply that the numbers will be prime. For example: 4 and 9 are relatively prime because they have no factor in common but they are not individually primes. But if two numbers are prime, then they must be relatively prime as well.