A b mod m. On our homework assignment, one of the questions is to prove the following (ab) mod m = [(a mod m Jul 11, 2025 · Your All-in-One Learning Portal: GeeksforGeeks is a comprehensive educational platform that empowers learners across domains-spanning computer science and programming, school education, upskilling, commerce, software tools, competitive exams, and more. Jun 8, 2015 · I have to compute efficiently a^^b mod m for large values of a,b,m<2^32 where ^^ is the tetration operator: 2^^4=2^(2^(2^2)) m is not a prime number and not a power of ten. Let \ (m\) \ (\in\) \ (\mathbb {Z_+}\). Rather, a ≡ b (mod m) asserts that a and b have the same remainder when divided by m. In simple terms, to compute (a/b) % M we instead calculate (a × b-1 ) % M where b-1 is the modular inverse of b modulo M ( i. e. a and b congruent modulo m, written only if a – b is divisible by m are said to be ≡ b (mod m), There is a small mistake in your explanation. Upvoting indicates when questions and answers are useful. Learn how it works with addition, subtraction, multiplication, and division using rules. Jul 12, 2025 · Q is the quotient B is the divisor R is the remainder The modulo operator (mod) helps us focus on the remainder: A mod B=R Example: 1 3 5 = 2 r e m a i n d e r 3 513 = 2 remainder 3 13mod 5=3 So, dividing 13 by 5 gives a remainder of 3 What is Modular Arithmetic? In modular arithmetic, numbers are reduced within a certain range, defined by the The congruence relation may be rewritten as a = k m + b, explicitly showing its relationship with Euclidean division. I'm taking a cryptography class and we are going over proofs and modulus equations for hashing. , b × b-1 ≡ 1 mod M ). The notation a b (mod m) means that m Jul 14, 2025 · Instead, division is performed by multiplying the dividend by the modular multiplicative inverse of the divisor under a given modulus. The "mod m" in a ≡ b (mod m) is a note on the side of the equation indicating what we mean when we say " ≡ " Fact: These two uses of "mod" are quite related: a ≡ b (mod m) if and only if a mod m = b mod m. ((a×b) mod M) I am learning about modular extraction and to efficiently calculate (a**b)%c. In fact there are five different definitions on the Wikipedia page alone! Although I would say that the statement you're trying to prove is practically part of the definition Jul 12, 2025 · Given three integers a, b, and M, where M is the modulus. Why is the formula (ab) (mod m) = (a (b (mod m))) (mod m) true? Although I am familiar with the concept of the modulus operator, I have little experience in modular algebra. Compute the result of the modular multiplication of a and b under modulo M. That is, a = p m + r, b = q m + r, where 0 ≤ r < m is the common Relation between ”x ≡ b mod m” and ”x = b MOD m” ny solutions for x while x = b MOD m is an EQUALITY. . modular exponentiation mod 7modular exponentiation mod 7 Mar 5, 2015 · You'll need to complete a few actions and gain 15 reputation points before being able to upvote. Can you help? I've defined the mod m o d operator to always return a value between 0 0 and m − 1 m 1, and noted that this may not match the % operator in some programming languages (see "terminology and notation" section). I am confused by the wikipedia article explaining the algorithm I found. So one can May 24, 2024 · What is modular arithmetic with examples. The letters m; n represent positive integers. Actually, you need to multiply A by B in power of f (P)-1 and f (P) for prime number is P-1. \ (a\) is congruent to \ (b\) modulo \ (m\) denoted as \ ( a \equiv b (mod \, n) \), if \ (a\) and \ (b\) have the remainder when they are divided by \ (n\), for \ (a, b \in \mathbb {Z}\). Oct 23, 2021 · A good first step is to recall the formal definition of a mod m a mod m -- more specifically, to recall the formal definition previously given by whoever stated the theorem, because that will give specific steps to take. BASIC PROPERTIES OF CONGRUENCES The letters a; b; c; d; k represent integers. Implementation to match this specification is left up to the reader. What's reputation and how do I get it? Instead, you can save this post to reference later. However, the b here need not be the remainder in the division of a by m. The following articles are prerequisites for this. Modular Arithmetic Remember: a ≡ b (mod m) means a and b have the same remainder when divided by m. Doesn't include zero a, b ∈ Z, m ∈ N. egsbdn qylnos eridy ktatzgzd qwfl mjbcix eii kyiyo rstc qxv