RELATIVELY PRIME NUMBERS

Two numbers are considered to be relatively prime, if there are no common factors other than 1. This means, no  other integer can divide both the numbers exactly.

In other words, if the greatest common factor (GCF) of two numbers is 1, then the numbers are relatively prime.

or

If two numbers do not have common divisor other than 1, they are said to be relatively prime.

Here, the two numbers can both be primes as (3, 11) or both can be composites as (16, 35) or one can be a prime and other a composite as (7, 14).

We can extend this concept for than two numbers also.

Questions 1-9 : Find which of the following pairs of numbers are relatively prime.

Question 1 :

4 and 12

Answer :

The above two numbers have the common divisors other than 1. They are 2 and 4.

So, 4 and 12 are not relatively prime.

Question 2 :

7 and 43

Answer :

The above two numbers have no common divisor other than 1.

So, 7 and 43 are relatively prime.

Question 3 :

5 and 3

Answer :

The above two numbers have no common divisor other than 1.

So, 5 and 3 are co-primes.

Question 4 :

8 and 17

Answer :

The above two numbers have no common divisor other than 1.

So, 8 and 17 are co-primes.

Question 5 :

8 and 15

Answer :

The above two numbers have no common divisor other than 1.

So, 8 and 15 are co-primes.

Question 6 :

14 and 21

Answer :

The above two numbers have common divisor other than 1. That is 7.

So, 14 and 21 are not co-primes.

Question 7 :

2 and 4

Answer :

The above two numbers have common divisor other than 1. That is 2.

So, 2 and 4 are not co-primes.

Question 8 :

1 and 2

Answer :

The above two numbers have no common divisor other than 1.

So, 1 and 2 are co-primes.

Question 9 :

8, 15 and 49

Answer :

Resolve 8, 15 and 49 into their prime factors.

8 = 2 x 2 x 2

15 = 3 x 5

49 = 7 x 7

There is no common factor or divisor for the numbers 8, 15 and 49.

So, the numbers 8, 15 and 49 are relatively prime.

Question 10 :

If two numbers are relatively prime, then, which is of the following must be true about the two numbers? Explain.

(A) Both the numbers must be prime

(B) One must be prime and other must be composite

(C) Both the numbers must be composite

(D) Both of them can be any numbers

Answer :

The correct answer choice is (D).

That is, both of them can be any numbers.

Consider the following pairs of relatively prime numbers.

(2, 3) ----> both of them are prime

(4, 5) ----> one is composite and other one is prime

(8, 15) ----> both of them are composite

From the above examples, it is clear that if two two numbers are relatively prime, they can be any numbers.

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