Check if two numbers are relatively prime (coprime) with our free calculator. Determine if their greatest common divisor (GCD) equals 1.
Two integers are relatively prime, also called coprime, when their greatest common divisor (GCD) equals exactly 1. This means the numbers share no common factors other than 1. For example, 8 and 15 are relatively prime because their only common divisor is 1, even though 8 = 2³ and 15 = 3 × 5 share no prime factors. The concept extends to any pair of positive integers and is fundamental in number theory, cryptography, and abstract algebra. Our Relatively Prime Calculator instantly determines whether two numbers are coprime by calculating their GCD, making it invaluable for students studying number theory, cryptographers developing encryption algorithms, and mathematicians exploring Diophantine equations.
The mathematical significance of relatively prime numbers extends far beyond simple classification. Euler's totient function φ(n), which counts integers less than n that are coprime to n, forms the foundation of RSA encryption and other modern cryptographic systems. Two numbers are coprime if and only if they have no prime factor in common—a property that creates fascinating patterns when examining large sets of numbers. For instance, the probability that two randomly chosen integers are coprime approaches 6/π² ≈ 0.6079, a remarkable result known as Euler's totient asymptotic density. This property underpins advanced mathematics, number theory algorithms, and computational cryptography. Understanding which numbers are relatively prime helps solve linear Diophantine equations and proves essential for modular arithmetic applications.
Practical applications of relatively prime numbers appear throughout mathematics and engineering. In fraction reduction, finding coprime pairs helps identify lowest terms (GCD = 1 means already reduced). In signal processing, relatively prime sampling rates prevent aliasing artifacts. In combinatorics and probability, counting coprime pairs within ranges requires understanding their distribution. Mechanical engineering uses relatively prime gear tooth counts to ensure smooth, even wear patterns. Music theory employs coprime integer ratios for harmonic intervals. Network routing algorithms and error-correcting codes depend on identifying coprime pairs for optimal performance. Whether solving number theory puzzles, implementing cryptographic protocols, or optimizing engineering systems, determining which numbers are relatively prime provides essential insights into the structure and behavior of integers.
Two numbers are relatively prime (or coprime) when their greatest common divisor (GCD) equals 1. This means they share no common factors except 1. For example, 9 and 16 are relatively prime because GCD(9, 16) = 1, even though 9 = 3² and 16 = 2⁴. They don't share any prime factors, so 1 is their only common divisor.
The most efficient method is the Euclidean algorithm: repeatedly divide the larger number by the smaller, replacing the larger with the smaller and the smaller with the remainder, until the remainder is 0. The last non-zero remainder is the GCD. For example, GCD(48, 18): 48 ÷ 18 = 2 remainder 12, then 18 ÷ 12 = 1 remainder 6, then 12 ÷ 6 = 2 remainder 0. So GCD(48, 18) = 6, meaning they're not coprime.
Yes, always! Any two consecutive integers n and (n+1) are always relatively prime. This is because any common divisor of consecutive integers would also divide their difference: (n+1) - n = 1. Since only 1 divides 1, their GCD must equal 1. This property makes consecutive integers extremely useful in various mathematical proofs and applications.
Relatively prime numbers are foundational to RSA encryption and public-key cryptography. In RSA, two large prime numbers are multiplied to create a public modulus n. The totient φ(n) = (p-1)(q-1) consists of numbers coprime to n, which enables key generation and encryption/decryption operations. The security of RSA depends on the difficulty of factoring n into its prime components, with coprime number properties ensuring mathematical validity.
By mathematical convention, relatively prime numbers are typically defined for positive integers only. However, the GCD can be extended to negative integers, where GCD(a, b) = GCD(|a|, |b|), meaning we consider absolute values. Zero is never relatively prime to any non-zero number because GCD(0, n) = |n|, which equals 1 only when n = ±1. So technically GCD(0, 1) = 1, but the concept of 'relatively prime' applies to positive integer pairs.