Number Theory is the study of integers and their properties — it is the mathematical foundation of modern cryptography, coding theory, algorithm analysis, and random number generation. From RSA encryption that secures internet communications to primality testing that underpins cryptographic keys, from GCD algorithms to modular arithmetic in hashing, number theory is essential for cybersecurity, data science, and algorithm design.

1. Divisibility and Prime Numbers

A number a divides b (a | b) if there exists integer k such that b = ak. Key divisibility rules: by 2 (last digit even), 3 (sum of digits divisible by 3), 4 (last two digits divisible by 4), 5 (last digit 0 or 5), 6 (divisible by 2 and 3), 9 (sum of digits divisible by 9), 10 (last digit 0). Prime numbers are integers > 1 with exactly two divisors (1 and itself). The Fundamental Theorem of Arithmetic states every integer > 1 factors uniquely into primes.

2. Greatest Common Divisor and Least Common Multiple

GCD(a,b) is the largest integer dividing both a and b. Euclid's algorithm computes GCD efficiently: gcd(a,b) = gcd(b, a mod b). Properties: gcd(a,b) × lcm(a,b) = a × b. The Euclidean algorithm is fundamental to many cryptographic operations.

3. Modular Arithmetic and Congruences

a ≡ b (mod m) means m divides (a-b). Modular arithmetic is the mathematics of remainders and is the foundation of cryptography, hashing, and random number generation. Fermat's Little Theorem: a^(p-1) ≡ 1 (mod p) for prime p. Euler's theorem: a^(φ(n)) ≡ 1 (mod n) when gcd(a,n)=1.

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