# Co-prime numbers

**Co-prime numbers**, also known as relatively prime numbers, are pairs of numbers that share no common divisors other than 1. For instance, the pairs (4, 7), (5, 9), and (11, 13) are all co-prime because the only factor they have in common is 1. Interestingly, co-prime numbers do not necessarily have to be prime themselves.

## What Are Co-Prime Numbers?

Co-prime numbers are defined as a pair of integers where the greatest common divisor (GCD) is 1. This means that there is no number other than 1 that divides both numbers. A common term used for these numbers is "mutually prime" or "relatively prime."

### Examples of Co-Prime Number Pairs

Here are some examples of co-prime pairs:

(2, 3)

(3, 5)

(4, 9)

(5, 7)

(11, 13)

(17, 19)

### How to Identify Co-Prime Numbers

To determine if two numbers are co-prime, you can use the concept of their greatest common divisor (GCD). For instance, consider the numbers 5 and 9. The factors of 5 are 1 and 5, while the factors of 9 are 1, 3, and 9. The only common factor between 5 and 9 is 1, which means (5, 9) forms a co-prime pair.

### Key Properties of Co-Prime Numbers

Understanding co-prime numbers involves several important properties:

**1. Product of Co-Prime Numbers**

When two co-prime numbers are multiplied, their highest common factor (HCF) is always 1. For example, the numbers 5 and 9 are co-prime, and the HCF of their product (45) and either number remains 1.

**2. Least Common Multiple (LCM)**

The LCM of two co-prime numbers is simply their product. For instance, for the numbers 5 and 9, the LCM is

5 × 9 = 45

**3. Pairing with 1**

Any number paired with 1 forms a co-prime pair because 1 is co-prime with every integer.

**4. Even Numbers and Co-Prime Pairs**

Two even numbers can never be co-prime because their common divisor is always at least 2.

**5. Sum of Co-Prime Numbers**

When two co-prime numbers are added, the sum is not necessarily co-prime with their product. For example, while 5 and 9 are co-prime, 5 + 9 = 14 is not co-prime with 5 × 9 = 45.

**6. Prime Numbers**

Two distinct prime numbers are always co-prime. For example, 29 and 31 are both prime, and their only common factor is 1.

**7. Consecutive Numbers**

Two consecutive integers are always co-prime. For example, 14 and 15 are consecutive numbers with no common factors other than 1.

There are multiple such combinations where 1 is the only common factor. From 1 to 100, co-prime number pairings include (1, 2), (3, 67), (2, 7), (99, 100), (34, 79), (54, 67), (10, 11), and so on.

### Examples of Co-Prime Numbers from 1 to 100

Here are some co-prime pairs within the range from 1 to 100:

- (1, 2)
- (3, 67)
- (2, 7)
- (99, 100)
- (34, 79)
- (54, 67)
- (10, 11)

### Conclusion

Co-prime numbers are a fundamental concept in number theory with interesting properties and applications. Whether dealing with integers, primes, or consecutive numbers, understanding co-prime relationships enriches mathematical knowledge.

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## Frequently Asked Questions on Co-prime numbers

Co-prime numbers, also known as relatively prime numbers, are two numbers that have no common divisor other than 1. In other words, the greatest common divisor (GCD) of these numbers is 1. Co-prime numbers do not have any factors in common apart from the number 1.

**Examples of Co-Prime Numbers:**

**8 and 15:**The GCD of 8 and 15 is 1, so they are co-prime.**9 and 16:**The only common divisor between 9 and 16 is 1, making them co-prime.**14 and 25:**Since their GCD is 1, 14 and 25 are co-prime.

Between 1 and 100, there are many pairs of co-prime numbers. A co-prime number pair from 1 to 100 consists of two numbers that have no common divisor other than 1.

**Example Pairs of Co-Prime Numbers from 1 to 100:**

**3 and 4:**GCD(3, 4) = 1**5 and 9:**GCD(5, 9) = 1**21 and 22:**GCD(21, 22) = 1

These pairs are just a few examples; many such pairs exist within this range.

Yes, 7 and 13 are co-prime numbers. To determine this, you check their greatest common divisor (GCD). For 7 and 13, the GCD is 1 because they have no common factors other than 1. Thus, they are co-prime.

Yes, 18 and 25 are co-prime numbers. When you find the greatest common divisor of 18 and 25, it is 1. This means that 18 and 25 do not share any common factors other than 1, so they are considered co-prime.