# COPRIME NUMBERS (को प्राइम संख्या )

## What are Coprime Numbers:

We have already learnt about What is trigonometry and all basic trigonometric identities. Hope all of my dear readers understood all the concepts. Here, in this post, we are going to share again one of the most interesting content of Mathematics, the Coprime Numbers definition and complete  concepts – Coprime numbers definition, examples, algorithms etc in this one single post. So, Read the complete post to kill the confusion and objections regarding it. Lets start the coprime numbers definition

 Coprime Numbers : Two (or more) numbers are said to be coprime, if both of them are divisible by only 1 together. In other words, Two (or more) numbers are said to be Coprime, if their only common factor is 1.

For example, 4 and 9 are two different numbers. 4 is divisible by 2 & 4 but not 9, similarly 9 is divisible by 3 & 9 but not 4. That means, if we consider 4 and 9 together, they are divisible by 1 only. So, 4 and 9 are coprime numbers.

This is the definition of Coprime numbers. Study it.

If we take two numbers only, which are coprime, then it is called Pairwise Coprime. We can also assume more than 2 numbers together to form coprime numbers. For example, 2,5,7 ; 3, 7, 10, 17 are coprime, which are not Pairwise.

### Coprime Numbers meaning in hindi (को प्राइम संख्या / सह अभाज्य संख्या)

दो (या अधिक) संख्याओं को सह अभाज्य कहा जाता है, यदि दोनों एक साथ केवल 1 से विभाज्य हैं। दूसरे शब्दों में, दो (या अधिक) संख्याओं को कोप्राइम कहा जाता है, यदि उनका एकमात्र सामान्य भाजक 1 है। उदाहरण के लिए, 4 और 9 दो अलग-अलग संख्याएँ हैं। 4; 2 और 4 से विभाज्य है, लेकिन 9 नहीं, इसी तरह 9; 3 और 9 से विभाज्य है, लेकिन 4 नहीं। इसका मतलब है, अगर हम 4 और 9 को एक साथ मानते हैं, तो वे केवल 1 से विभाज्य हैं। तो, 4 और 9 कोप्रेम संख्या (सह अभाज्य) हैं।

यदि हम केवल दो संख्याएँ लेते हैं, जो कि सह अभाज्य हैं, तो इसे Pairwise Coprime कहा जाता है। हम सह अभाज्य नंबर बनाने के लिए 2 से अधिक संख्याओं को एक साथ मान सकते हैं। उदाहरण के लिए, 2,5,7; 3, 7, 10, 17 कोप्रेम हैं, जो कि पेयरवाइज नहीं हैं।

This is the meaning or coprime numbers definition in hindi.

## Coprime Number Examples

There are infinite number of examples for co-prime numbers. Some examples of co-prime numbers are: 1,2,3 ; 2,3 ; 4, 9 ; 2, 3 5 etc. Examining the natural numbers and integers, we can say that, all the pairs of consecutive natural numbers are coprime numbers. For examples, 5, 6 ; 19,20 ; 38,39 ; 78,79 ; 100,101 ; 203; 204 ; 1105,1106 are co-prime numbers. Any negative number, other than -1, cannot create a pair of co-prime number with any number. For example: -1, -1 ; -7,-10 etc are not prime.

### Coprime Number Examples in Hindi

सह-अभाज्य संख्याओं के लिए अनंत संख्या के उदाहरण हैं। सह-अभाज्य संख्याओं के कुछ उदाहरण हैं: 1,2,3; 2,3; 4, 9; 2, 3 5 आदि। प्राकृतिक संख्याओं और पूर्णांक की जांच करते हुए, हम कह सकते हैं कि, लगातार प्राकृतिक संख्याओं के सभी जोड़े कोप्रेम संख्या हैं। उदाहरण के लिए, 5, 6; 19,20; 38,39; 78,79; 100,101; 203; 204; 1105,1106 आदि सह-प्राइम संख्या हैं। -1 के अलावा, कोई भी ऋणात्मक संख्या किसी भी संख्या के साथ सह-अभाज्य संख्या की एक जोड़ी नहीं बना सकती है। उदाहरण के लिए: -1, -1; -7, -10 आदि प्राइम नहीं हैं।

## Coprime Numbers Properties

Some basic Properties of co-prime numbers are given bellow-

• 1 is the only common factor of co-prime numbers.
• HCF or GCD of any co-prime numbers is 1
• “1 and -1” can create a pair of coprime number with any natural number.
• Two prime numbers always create a pair of co-prime numbers.
• Two different numbers, which are not prime, can also create co-prime numbers. (example – 4 and 9)
• pair of Consecutive natural numbers are always co-prime.
• Negative numbers cannot be co-prime other than -1.
• 0 cannot create co-prime numbers with any number.
• Two same numbers cannot be co-prime other than (1, 1)
• For any two coprime number a and b, we can get another two integers x and y such that ax + by = 1

## Coprime Number from 1 to 100

It is very difficult to write all the pair of co-prime numbers which are in between 1 and 100. Any number with 1 will create a pair of co-prime numbers, So, with 1, we can get total 100 pairs of co-prime numbers. In addition to this, all the consecutive pairs from 1 to 100 are co-prime. Thus we can get 99 pairs of consecutive natural number from 1 to 100. After this, 2,5 ; 2, 7; 2; 9; 2; 11; 2, 13 upto 2,99 are also co-prime starting with 2. Thus, 3, 5; 3, 7; 3, 10 upto 3, 100 are also coprime starting with 3. Thus, we can get many coprime numbers between 1 to 100.

सभी सह-अभाज्य संख्याओं की जोड़ी को लिखना बहुत मुश्किल है जो 1 और 100 के बीच में हैं। 1 के साथ कोई भी संख्या सह-अभाज्य संख्याओं की एक जोड़ी बनाएगी, इसलिए, 1 के साथ, हम सह-कुल 100 जोड़े सह-अभाज्य सँख्या प्राप्त कर सकते हैं। इसके अतिरिक्त, 1 से 100 तक की सभी लगातार जोड़ियाँ सह-अभाज्य सँख्या हैं। इस प्रकार हम 1 से 100 तक लगातार प्राकृतिक संख्या के 99 जोड़े प्राप्त कर सकते हैं। इसके बाद, 2,5; 2, 7; 2, 9; 2; 11; 2,13 …. 2,99 तक 2 से शुरू होने वाले कोप्राइम हैं। इस प्रकार, 3, 5; 3, 7; 3, 10 …. 3, 100 तक 3 से शुरू होने वाले कोप्राइम हैं। इस प्रकार, हम 1 से 100 के बीच कई कॉपीरिम नंबर प्राप्त कर सकते हैं।

## Relatively Prime OR Mutually Prime Numbers

Coprime numbers are also known as relatively prime or mutually prime numbers. In other words, Relatively Prime, Mutually prime and co-prime are of same meaning. All these means, a set of numbers, which have only one common divisor or factor, that is = 1

### Prime Numbers

Concepts of prime numbers and co-prime numbers are totally different. To be co-prime, we must need to have at least 2 different numbers. On the other hands, only one single number can be a prime number. Definition of Prime number is that, a number is said be prime, if it has exactly 2 positive factor. Number of factor must be two, not less than 2 or not greater than 2. Clearly, the two factors are 1 and itself.

अभाज्य संख्याओं और सह-अभाज्य संख्याओं का अर्थ बिलकुल अलग है। कोप्राइम होने के लिए, हमें कम से कम 2 अलग-अलग संख्याएँ होनी चाहिए। दूसरी ओर, केवल एक ही संख्या एक अभाज्य संख्या हो सकती है। एक संख्या एक अभाज्य संख्या है, अगर इसमें 2 सकारात्मक भाजक हैं। विभाजक की संख्या दो होनी चाहिए, दो से कम या दो से अधिक नहीं। स्पष्ट रूप से, दो कारक 1 और स्वयं हैं।

## Difference between prime and co-prime numbers

 Prime Number Coprime Numbers Only One Single Number Must be two or more together Must be exactly 2 factor Must be only one common factor which is 1 Negative numbers are not Prime -1 can create pairwise corpime with any natural number The largest known prime number is 282,589,933 − 1, a number which has 24,862,048 digits when written in base 10. Even Biggest Pairwise Coprime is unknown.

Some Special Knowledge For Prime Numbers:

• 1 is not Prime. Because 1 = 1 1 1 …. , but 1 has only 1 factor which is = 1 . This violets the property of prime number having exactly two factors.
• 2 is the least or smallest prime number which is also even. In other words, the only even prime number is = 2
• 3 is the smallest odd prime number.
• Negative integers are not prime. Because, Factors of a Prime numbers must be Positive
• The largest known prime number is 282,589,933 − 1, a number which has 24,862,048 digits when written in base 10.

## Biggest Coprime Numbers

The answer is null. As we can not determine the biggest ever natural number. The pair of biggest natural number and the number 1 less than it, will be the biggest pairwise co-prime numbers.

### Co-prime Numbers Algorithm

Algorithm 1: Let a and b be any two integers. a and b will be coprime if and only iff

ax + by = 1

Proof: Let, a and b are co-prime.

Then, GCD(a, b) =  1

⇒ ax + by = 1

Conversely, Let, ax + by = 1

Means, GCD(a, b) = 1

We know that, GCD of any pairwise co-prime number is 1, so, a and b are co-prime.

## USE OF COPRIME NUMBERS

One of the most mentionable use of Coprime Numbers is – while we representing a rational number, it is always expressed as $\frac p q$ where p and q are Coprime Numbers and q ≠ 0. The number will be ratiobal if and only if p and q are coprime, otherwise not.

सह-अभाज्य के सबसे उल्लेखनीय उपयोग में से एक है – जब हम एक परिमेय संख्या का प्रतिनिधित्व करते हैं, तो इसे हमेशा $\frac p q$ के रूप में व्यक्त किया जाता है, जहां p और q कोप्राइम नंबर हैं और q ≠ 0.

#### Q. 1 and 1 are coprime or NOT?

Answer: 1 and 1 are coprime as the only common factor of them is 1

#### Q. 1 and -1 are co-prime or not?

Answer: Yes, they are coprime as the only common factor is 1

#### Q. –1 and -1 are co-prime or not?

Answer: No, they are not coprime as the  common factors are 1 and -1 : two factors.

#### Q. Negative numbers are coprime or not?

Answer: -1 can be coprime with any natural number. However, two negative numbers cannot be coprime

#### Q. Two Composite number can be coprime or not?

Answer: Sometimes, Two composite number can be coprime. Example 4 and 9 ; 8 and 15 etc.