GCF Of Prime Numbers
The GCF of any two different prime numbers is always 1. In Grade 6 Saxon Math Course 1, students learn this rule because prime numbers have exactly two factors each (1 and themselves), so the only factor two different primes can share is 1. For example, GCF(5, 17) = 1 because factors of 5 are {1, 5} and factors of 17 are {1, 17}, with only 1 in common. When GCF = 1, two numbers are called relatively prime, a property used in fraction simplification.
Key Concepts
Property The greatest common factor (GCF) of any two different prime numbers is always 1.
Examples The GCF of 7 (factors: 1, 7) and 13 (factors: 1, 13) is $1$. The GCF of 3 (factors: 1, 3) and 19 (factors: 1, 19) is $1$. The GCF of 23 (factors: 1, 23) and 5 (factors: 1, 5) is $1$.
Explanation Prime numbers are famously picky, with only 1 and themselves as factors. So, when you compare two different primes, like 5 and 11, the only factor they have in common is always the number 1. This makes finding their GCF super easy—it’s the one and only common factor they could possibly share!
Common Questions
What is the GCF of any two different prime numbers?
Always 1. Different primes share no factors other than 1.
Why is GCF(5, 17) = 1?
Factors of 5: {1, 5}. Factors of 17: {1, 17}. The only common factor is 1.
What does 'relatively prime' mean?
Two numbers are relatively prime (or coprime) when their GCF is 1 — they share no common factor other than 1.
Are 4 and 9 relatively prime even though neither is prime?
Yes. Factors of 4: {1, 2, 4}. Factors of 9: {1, 3, 9}. Only common factor is 1, so GCF = 1.
How does GCF = 1 affect fraction simplification?
If the numerator and denominator have GCF = 1, the fraction is already in lowest terms and cannot be simplified further.