In This Lesson /jzgM8oVWMRMIMp5eAdYR+Nvu7sMN4cBK0zUKVJWfGGhoW5jHGncCkr7562K/oxDovA9miyE4DqFKswojvq9k8BE6FBxlOmofzrAJfOzmj8Oh6VuicA42D3lKTkuhdhWCQwabwJHb8oT0OGC6AdIGMX9vs9DjYTYBzAjOWjVQC9sIJJjo9Vw+8rQG2OnNjpzHE54iTgqO4QfG6PLl/W6rdrXwkEb1X7nGESNNEQdfDmO7htPJeI7x8BsphIvKWynYxrEgE39aVCaOpy5h5nK31whh4IxlOW/2L+vv3TwlZBkxRgBKTL3aAXzd2Uw2uhc4OeHPlunwJXHZoDyTygXJVxd/DI5gnGsHNABkQykq5u/x6NIYM++VAQ5JXt7A97G9gVWWFN7f6Kba37tk26zqmNWu74stHcza5+e+SJNL+9uCCjDMBNySaoFopC9gA9Rl1QMCxwG5Fu56lQFkZcmljWG65oMQE1IW04mkh4LPRlSiccyCm8wgEe8tnrmefeRyAjuboKtPoLVsvB2cU7/aVwh2hMmslyKySNoq5oLWEBYe87j3AEWAhWIpzhFFXKMH6ykhZQd1PNDd6QYm23P6vfzHdeIPqezlrtNJlsgA2O8DOZI7FcsXuLsEA0eBtzM2PSCpS1meqPZZp9bCqR4fgT2ICkUx7e6vtF1OoYVzvXDqQm8LYb04hCkR+LB9Yz9P9vnpSpd1r0g6Id7rgMxHZInAAeDfc7SObaweVAcMkKGBEtFk+TH Congruences 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 Modular Operations Modular Inverse Fermat & Euler Chinese Remainder Theorem Congruences
a ≡ b (mod m) means m divides (a − b)
Equivalently: a mod m = b mod m
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(a + b) mod m = ((a mod m) + (b mod m)) mod m
(a · b) mod m = ((a mod m) · (b mod m)) mod m
aⁿ mod m → use repeated squaring (fast exponentiation)
Example: 7¹³ mod 11 7¹ = 7, 7² = 49 ≡ 5, 7⁴ ≡ 5² = 25 ≡ 3, 7⁸ ≡ 3² = 9
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This is essential for division in modular arithmetic and for RSA decryption .
Fermat & Euler
Fermat's Little Theorem: aᵖ⁻¹ ≡ 1 (mod p) if p is prime and gcd(a,p) = 1
Euler's Theorem: a^φ(n) ≡ 1 (mod n) if gcd(a,n) = 1
Euler's totient: φ(n) = n · ∏(1 − 1/p) for each prime p dividing n
Fermat's theorem is a special case of Euler's (since φ(p) = p − 1). These are the theoretical backbone of RSA encryption . The exponential functions connect to the structure of multiplicative groups mod n.
Chinese Remainder Theorem
If m₁, m₂, …, mₖ are pairwise coprime, then:
x ≡ a₁ (mod m₁), x ≡ a₂ (mod m₂), …, x ≡ aₖ (mod mₖ)
has a unique solution mod (m₁·m₂·…·mₖ)
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CRT says you can reconstruct a number from its remainders — like reassembling a puzzle from pieces. This has applications in computer science (parallel computation),
cryptography (speeding up RSA), and even calendar calculations.