Python One-Liner: Compute GCD (Greatest Common Divisor) of Two Numbers!
Автор: CodeVisium
Загружено: 2025-04-18
Просмотров: 2227
Calculating the greatest common divisor (GCD)—the largest positive integer that divides two numbers without remainder—is essential in number theory, cryptography, and many algorithmic problems
Why the GCD of 360 and 48 Is 24
Euclidean Algorithm Steps
Remainder Sequence:
360 % 48 → 24
48 % 24 → 0
Once the remainder is zero, the nonzero divisor is the GCD. Therefore, GCD(360, 48) = 24
Prime Factorization Method
360 = 2³ × 3² × 5
48 = 2⁴ × 3
Common primes with lowest exponents → 2³ × 3 = 8 × 3 = 24
Long Way: Inline Euclidean Algorithm
Normalization:
Convert inputs to absolute values so the algorithm handles negative numbers correctly
Iterative Loop:
The loop while b: applies the identity
gcd(a,b)=gcd(b,amodb)
reducing the problem size each iteration until b becomes zero
Result:
When b is zero, a holds the GCD. For 360 and 48, this yields 24
One‑Liner: math.gcd()
Built‑in Convenience:
Python’s math.gcd(*integers) (available since Python 3.5) provides a single‑call solution that is highly optimized in C and handles edge cases (like zeros and negatives) gracefully
Usage Example:
import math
math.gcd(360, 48) # Returns 24
Advantages:
Concise: One line vs. several.
Performant: Lower‑level implementation for speed.
Robust: Correctly handles gcd(0, 0) == 0 and negative inputs
Practical Applications
Cryptography: Ensuring keys are co‑prime (GCD = 1) in RSA and other public‑key systems
Simplifying Fractions: Dividing numerator and denominator by their GCD for lowest terms.
Algorithm Design: Building blocks for LCM computation (lcm(a,b)=a*b//gcd(a,b)) and solving Diophantine equations (extended GCD) .
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Codes:
Long Way: Inline Euclidean algorithm
a, b = 360, 48
a, b = abs(a), abs(b)
Ensure non‑negative inputs
while b:
Repeat until b becomes zero
a, b = b, a % b
Apply (a, b) ← (b, a mod b)
print(a)
'a' now holds GCD: 24
One‑Liner: Using Python's built‑in math.gcd()
import math
Import Python’s math module
print(math.gcd(360, 48))
Compute and print GCD in one call: 24
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