Co-Prime Numbers

Grade 6 Math Worksheets

Co-Prime Numbers

Co-prime numbers are integers that have no common positive integer factors other than 1. In other words, the greatest common divisor (GCD) of co-prime numbers is equal to 1.

Here are some key points about co-prime numbers:

Definition

Two numbers a and b are coprime if gcd(a, b)=1, where gcd(a,b) represents the greatest common divisor of a and b.

Examples

8 and 15 are coprime because the only positive integer that divides both of them is 1.

12 and 35 are coprime because gcd(12,35)=1

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Co-Prime Numbers - Grade 6 Math Worksheet PDF

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How to Find Co-prime Numbers?

Finding co-prime numbers involves determining whether two given numbers share any common factors other than 1. Here’s a step-by-step approach to finding co-prime numbers:

Choose Two Numbers

1. Select two integers for which you want to determine whether they are co-prime.

2. Find the prime factorization of each number.

If the two numbers do not share any common prime factors, they are co-prime.

Example 1:

Let us consider two numbers 21 and 40.

Prime factorization of 21: 3×7
Prime factorization of 40: 23×5

Since they don’t share common prime factors, 21 and 40 are co-prime.

Example 2:

Let us consider two numbers 5 and 9.

Prime Factorization of 5: The number 5 is a prime number, and its prime factorization is simply 5 itself.
Prime Factorization of 9: The prime factorization of 9 is 3×3

Now, let’s check if they share any common prime factors:

5 has the prime factorization: 5

9 has the prime factorization: 3×3

Since 5 and 9 do not share any common prime factors, they are co-prime numbers. The absence of common factors other than 1 makes them relatively prime to each other.

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Properties of Co-prime Numbers

Property 1

The most fundamental property is that the greatest common divisor of co-prime numbers is 1. Mathematically, if GCD(a,b)=1, then a and b are co-prime.

Property 2: No Common Factors

Co-prime numbers have no common prime factors. For example, 12 and 35 are co-prime because the prime factorization of 12 is 2×2×3 and the prime factorization of 35 is 5×7 and there are no common prime factors.

Property 3: Infinite Pairs

There are infinitely many pairs of co-prime numbers. This is a consequence of the fact that there are infinitely many prime numbers, and any two distinct prime numbers are co-prime.

Property 4

The HCF of two co-prime numbers is always 1. This is because there are no common factors other than 1.

Property 5

The LCM of two co-prime numbers is the product of the two numbers. This is a consequence of the fact that when two numbers have no common factors other than 1, their LCM is the product of the numbers.

Co Prime Numbers and Twin Prime Numbers

Co-prime numbers and twin prime numbers are two different concepts in number theory.

Co-prime Numbers

Co-prime numbers (or relatively prime numbers) are two numbers that have no common factors other than 1. In other words, their greatest common divisor (GCD) is 1. For example, (6, 25) is a pair of co-prime numbers because the only positive integer that divides both 6 and 25 is 1.

Twin Prime Numbers

Twin prime numbers are pairs of prime numbers that have a difference of 2. Examples include (3, 5), (11, 13), and (17, 19). The distinctive feature of twin primes is that there is exactly one even number (2) between them.

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Co-Prime Numbers FAQS

What are co-prime numbers?

Co-prime numbers, also known as relatively prime or mutually prime numbers, are two numbers that have no common positive integer other than 1 as a divisor. In other words, their greatest common divisor (GCD) is 1.

How can you determine if two numbers are co-prime?

Calculate the greatest common divisor (GCD) of the two numbers. If the GCD is 1, then the numbers are co-prime. If the GCD is greater than 1, then they are not co-prime.

Can more than two numbers be co-prime?

Yes, more than two numbers can be co-prime if the GCD of all possible pairs of numbers is 1. For example, the numbers 8, 15, and 27 are co-prime because the GCD of (8,15), (8,27), and (15,27) is 1.

Are all prime numbers co-prime to each other?

Yes, all prime numbers are co-prime to each other. This is because the only positive integer that divides any pair of distinct prime numbers is 1.

Are there infinitely many pairs of co-prime numbers?

Yes, there are infinitely many pairs of co-prime numbers. This is a consequence of the infinitude of prime numbers, as any two distinct prime numbers form a pair of co-prime numbers.

Can co-prime numbers be negative?

Yes, co-prime numbers can be negative. The concept of co-primality focuses on the common factors, and whether the numbers are positive or negative doesn’t affect the definition.

Gloria Mathew writes on math topics for K-12. A trained writer and communicator, she makes math accessible and understandable to students at all levels. Her ability to explain complex math concepts with easy to understand examples helps students master math. LinkedIn

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