Core · Module 12 · Lesson 1 of 4
Euclidean GCD, Modular Arithmetic, and Binary Exponentiation
Compute greatest common divisors in logarithmic steps, prevent 64-bit overflow with modular arithmetic, and raise numbers to huge powers in O(log b) multiplications.
By the end you can
- Prove and trace Euclid's GCD algorithm gcd(a, b) = gcd(b, a mod b) in O(log(min(a, b))) steps
- Apply modular addition, subtraction, and multiplication rules without signed 64-bit overflow
- Compute a^b mod m in O(log b) time using binary exponentiation and modular inverse via Fermat's Little Theorem
Builds on Bits, bytes and number representation, Counting steps.
01 · A question first
Computing 3^(10^9) mod 1,000,000,007 with a simple loop takes a billion multiplications and times out; binary exponentiation finishes in just 30 multiplications. How does reading the binary bits of the exponent skip 99.999997% of the work?
Make a guess before you read on.
02 · The idea
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What's inside
- Intuitive Explanation & Real-World Analogy
- Core Concepts & Visual Walkthrough
- Pseudocode
- Multi-Language Implementations (C++, Java, Python)
- Time & Auxiliary Space Complexity Derivation
- Edge Cases & Common Beginner Mistakes
- Alternative Approaches (Brute Force vs. Optimal)
- Pattern Recognition Guide & Three-Stage Practice Plan
03 · Practice
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