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Relatively Prime

Relatively Prime

Describes two numbers for which the only common factor is 1. In other words, relatively prime numbers have a greatest common factor (gcf) of 1. For example, 6 and 35 are relatively prime (gcf = 1). The numers 6 and 8 are not relatively prime (gcf = 2).

Note: When you are asked to write a fraction as a ratio of relatively prime integers, you are being asked to simplify the fraction as much as possible.

 

See also

Composite numbers, least common multiple

Key Formula

gcd⁡(a,b)=1\gcd(a, b) = 1
Where:
  • aa = The first integer
  • bb = The second integer
  • gcd⁡\gcd = Greatest common divisor (same as greatest common factor)

Worked Example

Problem: Determine whether 28 and 45 are relatively prime.
Step 1: Find the prime factorization of each number.
28=22×745=32×528 = 2^2 \times 7 \qquad 45 = 3^2 \times 5
Step 2: Identify any shared prime factors. The primes of 28 are 2 and 7. The primes of 45 are 3 and 5. There are no primes in common.
Step 3: Since the two numbers share no prime factors, their greatest common factor is 1.
gcd⁡(28,45)=1\gcd(28, 45) = 1
Answer: 28 and 45 are relatively prime because their GCF is 1.

Another Example

Problem: Determine whether 24 and 54 are relatively prime.
Step 1: Find the prime factorization of each number.
24=23×354=2×3324 = 2^3 \times 3 \qquad 54 = 2 \times 3^3
Step 2: Both factorizations contain the primes 2 and 3, so these numbers share common factors.
Step 3: The greatest common factor is the product of the lowest powers of the shared primes.
gcd⁡(24,54)=21×31=6\gcd(24, 54) = 2^1 \times 3^1 = 6
Answer: 24 and 54 are NOT relatively prime because their GCF is 6, not 1.

Frequently Asked Questions

Do two numbers have to be prime to be relatively prime?
No. Neither number needs to be a prime number. For example, 8 and 15 are both composite, yet they are relatively prime because their only common factor is 1. The word "prime" in "relatively prime" refers to the relationship between the two numbers, not to whether each number is individually prime.
Can a number be relatively prime to itself?
Only the number 1 is relatively prime to itself, since gcd⁡(1,1)=1\gcd(1,1) = 1. For any integer n>1n > 1, gcd⁡(n,n)=n≠1\gcd(n, n) = n \neq 1, so a number greater than 1 is never relatively prime to itself.

Relatively Prime (Coprime) vs. Prime Number

A prime number is a single integer greater than 1 with exactly two factors: 1 and itself. Relatively prime is a relationship between two integers — it says they share no common factor other than 1. Two composite numbers like 9 and 14 can be relatively prime even though neither is a prime number. Conversely, two prime numbers like 3 and 3 are not relatively prime to each other because gcd⁡(3,3)=3\gcd(3,3) = 3.

Why It Matters

Simplifying fractions relies directly on this concept: a fraction is fully simplified when its numerator and denominator are relatively prime. For instance, 2454\frac{24}{54} simplifies to 49\frac{4}{9} because gcd⁡(4,9)=1\gcd(4,9) = 1. The idea also appears in advanced topics like modular arithmetic and cryptography, where algorithms such as RSA require choosing numbers that are coprime.

Common Mistakes

Mistake: Assuming both numbers must individually be prime for them to be relatively prime.
Correction: Relatively prime describes the relationship between two numbers. Composite numbers like 8 and 25 can be relatively prime (gcd⁡(8,25)=1\gcd(8,25)=1) because they share no prime factors.
Mistake: Checking only a few small factors instead of all prime factors.
Correction: To be sure, find the complete prime factorization of both numbers or use the Euclidean algorithm. For example, 21 and 35 are not relatively prime — both are divisible by 7 — but you might miss this if you only check 2, 3, and 5.

Related Terms