what is co prime Related: Chapter Notes - Playing with Numbers?

Question

what is co prime Related:  Chapter Notes - Playing with Numbers?

Answer

Co-Prime Numbers

Co-prime numbers are those numbers which do not have any common factor other than 1. In other words, if two numbers have only 1 as their common factor, then they are said to be co-prime or relatively prime. For example, 6 and 35 are co-prime because their only common factor is 1.

Properties of Co-Prime Numbers

- Co-prime numbers are always positive integers.
- The number 1 is always co-prime with any other number.
- If the product of two numbers is co-prime with a third number, then both the numbers are co-prime to that third number.

Examples of Co-Prime Numbers

- 7 and 9 are co-prime numbers because they do not have any common factors other than 1.
- 4 and 9 are not co-prime numbers because they have a common factor 1 and 3.
- 21 and 22 are co-prime numbers because they do not have any common factors other than 1.

Uses of Co-Prime Numbers

Co-prime numbers are used in various mathematical applications like cryptography, number theory, and in solving problems related to fractions. They are also used in generating random numbers in computer science.

Conclusion

Co-prime numbers are an important concept in mathematics. They help us understand the properties of numbers and their relationship with each other. They are used in various fields of mathematics and science for solving problems and generating random numbers.

Answer

Answer

A Co-prime number is a set of numbers or integers which have only 1 as their common factor i.e. their

highest common factor

(HCF) will be 1. Co-prime numbers are also known as relatively prime or mutually prime numbers. It is important that there should be two numbers in order to form co-primes.

How to Find Co-prime Numbers?

Consider a set of two numbers, if they have no positive integer that can divide both, other than 1, the pair of numbers is co-prime.

Example 1: 21 and 22

For 21 and 22:

The factors of 21 are 1, 3, 7, and 21.
The factors of 22 are 1, 2, 11, and 22.

Here 21 and 22 have only one common factor that is 1. Hence, their HCF is 1 and are co-prime.

Example 2: 21 and 27

For 21 and 27:

The factors of 21 are 1, 3, 7, and 21.
The factors of 27 are 1, 3, 9, and 27.

Here 21 and 27 have two common factors; they are 1 and 3. HCF is 3 and they are not co-prime.

Properties of Co-Prime Numbers

Some of the properties of co-prime numbers are as follows.

1 is co-prime with every number.
Any two
prime numbers
are co-prime to each other: As every prime number has only two factors 1 and the number itself, the only common factor of two prime numbers will be 1. For example, 2 and 3 are two prime numbers. Factors of 2 are 1, 2, and factors of 3 are 1, 3. The only common factor is 1 and hence they are co-prime.
Any two successive numbers/ integers are always co-prime: Take any consecutive numbers such as 2, 3, or 3, 4 or 5, 6, and so on; they have 1 as their HCF.
The sum of any two co-prime numbers are always co-prime with their product: 2 and 3 are co-prime and have 5 as their sum (2+3) and 6 as the product (2�3). Hence, 5 and 6 are co-prime to each other.
Two even numbers can never form a coprime pair as all the even numbers have a common factor as 2.
If two numbers have their unit digits as 0 and 5, then they are not coprime to each other. For example 10 and 15 are not coprime since their HCF is 5 (or divisible by 5).

Co prime numbers from 1 to 100

There are several pairs of co-primes from 1 to 100 which follow the above properties. Some of them are:
(13, 14)
(28, 57)
(1, 99)
(2, 97)
(46, 67)
(75, 41) and so on.
Also, we can write any number with the combination of 1 as a coprime pair such as (22, 1), (31, 1), (4, 1), (90, 1), (1, 100). In this way, many coprime numbers are defined from 1 to 100.

Example Question Related to Co-prime Numbers

Question 1: Check whether 13 and 31 are co-prime.

Solution:

13 and 31 are two prime numbers; therefore, they are co-prime to each other. (Property 2)

The factors of 13 are 1, 13 and the factors of 31 are 1, 31.

They have only 1 as their common factor. So, they are coprime numbers.

Question 2: Check whether 150 and 295 are coprimes.

Solution:

Given two number are: 150 and 295
150 and 295 are divisible by 5.
From the properties of coprime numbers, 150 and 295 are not coprime.
Alternatively,
150 = 2 � 3 � 5 � 5
295 = 5 � 59
HCF(150, 295) = 5 ≠ 1
Therefore, 150 and 295 are not coprime.

Frequently Asked Questions From Co-Prime Numbers

What are Co-prime Numbers?

Co-prime numbers or relatively prime numbers are those numbers that have their HCF (Highest Common Factor) as 1. In other words, two numbers are co-prime if they no common factor other than 1.

What is the difference between prime and Coprime numbers?

A prime number is defined as a number that has no factor other than 1 and itself. On the contrary, co-primes are considered in pairs and two numbers are co-prime if they have no common factors other than 1.

How do you Find the Co-prime of a Number?

To find the co-prime of a number, find the factors of the number first. Then, choose any number and find the factors of the chosen number. All the numbers which do not have any common factor other than 1 will be the co-prime of the given number.

Are 18 and 35 Coprime Numbers?

Yes, 18 and 35 are co-prime numbers. The factors of 18 are 1, 2, 3, 6, 9, and 18 while the factors of 35 are 1, 5, 7, and 35. Since the HCF is 1, they are coprime.

Is 1 Coprime to All numbers?

Yes, 1 is coprime to all the numbers. Since HCF of 1 and any number is 1 itself. Hence, by the definition of coprime numbers, 1 said to be coprime with all numbers.

Answer

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