2 mod 4 remainder. One of these operators is the modulo operator (%), which returns the remainder of dividing two numbers. The first result in our calculator uses, as stated above, the function floor () to calculate modulo as reproduced below: a mod b = a - b × floor (a/b) The modulo calculator finds the solution of an expression x mod y = r. Modular arithmetic can be used in addition, subtraction, multiplication, and exponentiation. For positive values, there is no difference between Euclidean and truncating division. Number Theory 4. If today is So what is a modulu or modulus? Put simply, modulo is the math operation of finding the remainder when you divide two numbers together. In this case, 4 goes into 3 zero times with a remainder of 3. This page starts with some simple examples. What number has a remainder of 2 when divided by 3, a remainder of 3 when divided by 5 and a remainder of 2 when divided by 7? There are a couple of methods… The modulo (or "modulus" or "mod") is the remainder after dividing one number by another. The modular calculator only takes dividend and divisor from the user to calculate the remainder after the division. Try now! Jun 1, 2025 · The modulus operation finds the remainder after dividing one number (the dividend) by another (the divisor). 19^24 mod 21 4. Here’s how you can apply modular operations in different scenarios: Use the construction in the proof of the Chinese remainder theorem to find all solutions to the system of congruences x ≡ 2 (mod 3), x ≡ 1 (mod 4), and x ≡ 3 (mod 5). Example 3. In its basic form, the Chinese remainder theorem will determine a number Congruences in Modular Arithmetic Two integers a a and b b, with b> 0 b> 0, are said to be congruent modulo n n if their difference a−b a b is divisible by a positive integer n n. A modulo calculator is an online tool that helps you find the remainder when one number (the dividend) is divided by another (the divisor). In other words, n≡0(mod4), or n≡1(mod4), or n≡ 2(mod4), or n≡3(mod4). b = mod(a,m) returns the remainder after division of a by m, where a is the dividend and m is the divisor. Modulo is the operation of finding the Remainder when you divide two numbers. [1] Sunzi's original formulation Chinese remainder theorem. But, however, what follows this error, $\bar a^2=\bar r^2$ is correct. Therefore the (distinct) elements of Z/nZ are Method-2: Successive substitution method `x-=2\ ("mod "5)-> (1)` `x-=3\ ("mod "7)-> (2)` `x-=10\ ("mod "11)-> (3)` from (1) `x-=2\ ("mod "5)` Hence `x=2+5a` Now from May 16, 2015 · When you see "modulo", especially if you are using a calculator, think of it as the remainder term when you do division. 2 revisited). Here we will explain what 4 mod 2 means and show how to calculate it. For example, if you want to know 17 mod 5, the calculator instantly shows that 17 divided by 5 leaves a remainder of 2. Oct 30, 2023 · The question 'How many is 2%4' is a **modulus **operation in mathematics. Perfect for time, scheduling, and programming tasks. Easy, fast, and perfect for students learning modular arithmetic! Modulo operator – Denoted by “mod” in mathematics and “%” in computer science, the modulo operator returns the remainder after the division of two integers. /m). 4 mod 2 is short for 4 modulo 2 and it can also be called 4 modulus 2. The divmod (a, b) function can also be used to Example of Modulo operator (%): 10 % 4 gives 2 because 10 divided by 4 leaves a remainder of 2. How do we find these solutions? Understand the Problem The question is asking for the result of the modulus operation, specifically what is the remainder when 2 is divided by 4. For instance, in (a), how is $0\leq r^2<4$. a ≡ b mod n a ≡ b mod n Another way to express this is: Two integers a a and b b (b> 0 b> 0) are congruent modulo n n (n> 0 n> 0) if they leave the same remainder when divided by n n. Modulo is about measuring what’s left. You may also enter the math expression containing other integers and the following modular arithmetic operations: + addition modulo p - subtraction modulo p * multiplication modulo p / division modulo p (available for all numbers if All About Modulo Calculator: Some math ideas are about measuring how much. The Chinese remainder theorem is a theorem which gives a unique solution to simultaneous linear congruences with coprime moduli. There is no need for us to express 7 and −3 using the division algorithm. It returns the remainder of dividing the left hand op Free Chinese Remainder Theorem Calculator - Given a set of modulo equations in the form: x ≡ a mod b x ≡ c mod d x ≡ e mod f the calculator will use the Chinese Remainder Theorem to find the lowest possible solution for x in each modulus equation. This mathematical operation is abbreviated as ‘mod’. 25, and if k k is greater than −6 6 the remainder is 0. Calculation To calculate 2 mod 4, follow these steps: Divide: Divide 2 by 4. . Modulo Operator as Used in our Calculator This operation (or function) rounds a value downwards to the nearest integer even if it is already negative. The highest multiple is $$4$$4 itself. Thus the remainder is 7. 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 While you still can simply enter an integer number to calculate its remainder of Euclidean division by a given modulus, this modulo calculator can do much more. 1); // returns number 2 floor (2. How to calculate modulus without a calculator? - Use long division to find the remainder. Jul 15, 2020 · It follows the Euclidean division rule, meaning the remainder always has the same sign as the divisor. Notice that 3 4 = 81 = 1 in mod 5. An example of this kind of systems is the following; find a number that leaves a remainder of 1 when divided by 2, a remainder of 2 when divided by three and a remainder of 3 when divided by 5. 2 7. 4 Quotient-Remainder Theorem In this lecture: qPart 1: Quotient-Remainder Theorem qPart 2: div and mod, and applications in real-life qPart 3: Representing Integers in Quotient-Remainder of Proof qPart 4: Absolute Value 3 Welcome to the modulo calculator! The most accurate online tool to calculate the modulo operations. What is 4 mod 2? The mod or modulus or modulo is the remainder after dividing one number by another. For example : 8 mod 3 = 2 (remainder) Calculator Use In the ‘a’ field, enter the dividend value, and in the ‘b’ field, enter the divisor value. The two main things which we need for division problems are dividend and divisor. Since 4 goes into 2 zero times, you end up with a remainder of 2. 4 mod 2 is 0. 7^106 mod 143 5. So if it is Sunday today, 100 days from now it will be Tuesday! Enter any number . For example, “5 mod 3 = 2” which means 2 is the remainder when you divide 5 by 3. This function is often called the modulo operation, which can be expressed as b = a - m. Subtract: Subtract this Feb 23, 2022 · One value is the quotient, and the other is the remainder. Download the practice workbook, modify data, find new results and exercise! Find out what is 700. The % symbol in Python is called the Modulo Operator. The reason your calculator says 113 modulo 120 = 113 is because 113 < 120, so it isn't doing any division. The remainder when $$4$$4 is divided by $$2$$2 is $$0$$0. The Chinese remainder theorem calculator is here to find the solution to a set of remainder equations (also called congruences). Multiply: Multiply the quotient by 4: 0 * 4 = 0. This means they have the same remainder when divided by n. Modulo is the name given to the remainder value. Syntax of Modulus Operator If x and y are integers, then the expression: x % y; pronounced as "x mod y". How do you calculate mod 4? 1 mod 4 equals 1, since 1/4 = 0 with a remainder of 1. Is 6 ≡ 11(mod5) 6 ≡ 11 (m o d 5)? Yes, because 6 6 and 11 11 both belong to the same congruent/residue class 1 1. 2); // returns Equivalent to 3 2022 = x in mod 5. 2. Modulo Calculator Welcome to the Modulo Calculator also known as the Mod Calculator or the Modulus Calculator. Example: 14 mod 12 equals 2 Because 14 12 = 1 with a remainder of 2 So 14 o'clock becomes 2 o'clock. 42^-1 mod 5 Use our Modulo Calculator to find the remainder when dividing two numbers. But Why? Because, the the remainder when $ a^2$ is divided by $4$ is the same as that when $ r^2$ is divided by $4$ is Linear Congruences, Chinese Remainder Theorem, Algorithms Recap - linear congruence ax ≡ b mod m has solution if and only if g = (a, m) divides b. Consider a number of the form $4q+2$, you'll have an issue. The modulo operation calculates the remainder when a number is divided by another. 4: Find the remainder when 71453 7 1453 is divided by 8. Modulo 1. Subtract this highest multiple from the dividend $$4$$4. Perfect for students, teachers, and anyone needing fast, accurate division results. Apply the same and find X for the given set of congruent equations using CRT. 4 3. To find 1 mod 4 using the modulus method, we first find Jul 5, 2024 · 9 easy and suitable uses of MOD function in Excel. Souare each of these and reduce modulo 4 to see what you get. Home Modulo Calculator Modulo Calculator Easily find the remainder when dividing two numbers with our modulo calculator. Jan 12, 2014 · The Chinese Remainder Theorem is a method to solve the following puzzle, posed by Sun Zi around the 4th Century AD. What Is Modulo? So what is a modulu or modulus? Put simply, modulo is the math operation of finding the remainder when you divide two numbers together. The mod function follows the convention that mod(a,0) returns a. Then a ≡ b (mod n) iff a and b leave the same remainder when divided by n. Section 4. There are lots of reasons why you would want to use modulo, including checking if a number is even or odd, counting something for a certain The modulo operation, which is also often referred to as the mod or modulus operation, identifies the remainder after a given number is divided by another number. 3 - The Chinese Remainder Theorem Exercise 4abc: Find all of the solutions to each system of linear congruences Apr 26, 2024 · What is the Chinese remainder theorem with the statement, formula, proof, and examples. This is useful in lots of different situations, and especially in programming. (Hence the justification of Gauss’ phrasearithmetic of remainders. 21 3 mod 4 is the remainder when 3 is divided by 4. 75 0. For example, 10 % 2 will be pronounced as " Ten mod Apr 17, 2022 · 7319 ≡ 2 7319 ≡ 2 (mod 9) and 20 ≡ 2 20 ≡ 2 (mod 9). Example: 100 mod 9 equals 1 Because 100/9 = 11 with a remainder of 1 Example: 14 mod 12 equals 2 Because 14/12 = 1 with a remainder of 2 12-hour time uses modulo 12 (14 o'clock becomes 2 o'clock) It is where we end up, not how many times around. We perform the calculation: $$4 - 4 = 0$$4−4 = 0 Conclude the result. Answer As there is a consistent pattern emerging and we know that 1453 1453 is odd, then 71453≡7(mod8) 7 1453 ≡ 7 (m o d 8). 25 0. The modulo division operator produces the remainder of an integer division which is also called the modulus of the operation. Syntax MOD (x, y) Mod calculator This online calculator performs modulo operation on integers and on doubles as well (because it just returns remainder of division) This online calculator performs modulo operations on two given numbers, dividend and divisor. Example Consider we take modulo 4. See step-by-step solutions, formulas, and examples to master division in maths. The modulo operation, symbolized as "mod", finds the remainder of division between two numbers. 27^400 mod 619 6. The modern approach to modular Apr 25, 2024 · To calculate a mod b, which is for positive numbers, simply find the remainder of a divided by b in a division problem. The floor function returns the remainder with the same sign as the divisor. Our next goal is to define an arithmetic with remainders, again without calculating quotients. *floor(a. In mathematics, the Chinese remainder theorem states that if one knows the remainders of the Euclidean division of an integer n by several integers, then one can determine uniquely the remainder of the division of n by the product of these integers, under the condition that the divisors are pairwise coprime (no two divisors share a common factor other than 1). 7 We can use the property of powers: Calculate the first few powers of 2 modulo 7: 21 mod 7 = 2 22 mod 7 = 4 23 mod 7 = 1 Notice that the powers of 2 modulo 7 repeat every 3 steps: 2 4 1 2 4 1 Since , we find that 2 22. Example (3. In particular, every a is congruent to its remainder when divided by n, and no two distinct remainders are congruent modulo n. Here we will explain what 2 mod 4 means and show how to calculate it. These 4 remainder classes are: [0]4,[1]4,[2]4,[3]4. Suppose n = 5, n = 5, then the possible remainders are 0, 1, 2, 3, 0, 1, 2, 3, and 4, 4, when we divide any integer by 5 5. You’ve seen it in real life, even if you didn’t know it. Jul 12, 2025 · In C or C++, the modulo operator (also known as the modulus operator), denoted by %, is an arithmetic operator. 2 mod 4 is short for 2 modulo 4 and it can also be called 2 modulus 4. On calculators, modulo is often calculated using the mod () function: mod (a, b) = r. Learn how to use it with applications. Jul 23, 2025 · When dividing by 4, the remainder helps determine whether a number is fully divisible by 4 or if a smaller remainder is left over. Dividing numbers and not sure about the quotient or remainder? This tool is used to find the quotient and remainder of a division of two whole numbers Dividend and Divisor given by Dividend/Divisor = Quotient + Remainder/Divisor. By simple algebra, if 5k + 13 5 k + 13 is divided by 4 4, then if k ≤ −6 k ≤ 6 the remainder is 0. This function is used in mathematics, where the result of the modulo operation is the remainder of the Euclidean division. 2 mod 4 with+steps. For example, you could use this calculator to find out what day of the week it is 100 weeks from now: 100 mod 7 equals 2. アルゴリズムの違いによる商 (赤線)と余り (緑線)のグラフ 剰余演算 （じょうよえんざん）は、ある数値（被除数、 英: dividend）を別の数値（除数、 英: divisor）で 除算 し、余り（剰余 、 英: remainder）を取得する演算である [1][2]。 モジュロ演算 （英: modulo operation / modulus operation）とも呼ばれる Theorem 2 Let n ∈ N and a, b ∈ Z. Experience swift and accurate results every time! Modulo Calculator A quick and easy calculator to solve modulo operations. Modulo is a way of finding the remainder when one number is divided by another. The Chinese Remainder Theorem gives us a tool to consider multiple such congruences simultaneously. Master modular arithmetic with our power mod calculator, perfect for calculations with exponents. 3^302 mod 5 3. But in Python, as well as most other programming languages, it means something different. 75. To find the modulus of a mod b, use mod calculator. Hint: n is in one of the 4 remainder classes modulo 4 . Find the Quotient: The quotient is 0 because 4 goes into 2 zero times. Jul 10, 2023 · (a) To find the remainders when and 4 are divided by 7, we will use modular arithmetic. It’s written as: a m o d n a mod n. So x = 3 2022 = 3 4*505+2 = (3 4 ) 505 * 3 2 = 1 505 * 9 = 9 = 4 in mod 5. 4 7 Using Fermat's Little Theorem, which states for a prime and a base not divisible by 2. 7 mod 3 = 1 Explanation: When 7 is divided by 3, the quotient is 2, and the remainder is 1. 7); // returns number 2 floor (-5. If today is Modulo Calculator Write the numbers x (dividend) and y (divisor), and the modulo calculator will apply the modulo operation to find the final remainder and the order of the division. This is what I've done to solve it: Since k k is congruent to 2 mod 4 2 mod 4, any number in the set {, −10, −6, −2, 2, 4, 18, 22,} {, 10, 6, 2, 2, 4, 18, 22,} will solve the original congruence. For instance, "5 mod 3" evaluates to 2, as 5 divided by 3 leaves a remainder of 2. It is used in finding even/odd numbers, cyclic patterns, and leap year calculations. That is to say when 6 6 and 11 11 are divided by 5 5 the remainder is 1. Given that the n i portions are not pairwise coprime and you entered two modulo equations, then the calculator will attempt to solve using the Dec 3, 2012 · 208 There is a difference between modulus (Euclidean division) and remainder (C's % operator). The % sign denotes a modulus (remainder) operation. 2); // returns Use this tool to find the remainder when a number (the dividend) is divided by another number (the divisor). This modulo operation helps to determine the remainder. 8. 4. If you are asking "what is 2 mod 4?" then what you really need to know is "what is the remainder when I divide 2 by 4?". n/ were congruent modulo 9 since they both had the same remainder when divided by 9. 2, we saw that n and s. 2 over 4. Understanding the Modulo Operation and Remainder Classes The modulo operation is written as "mod," this is important to remainder classes. Use our free Modulo Calculator to find modulus and remainder with step-by-step solutions. In case the advantage isn’t clear, we recall our earlier example. a ≡ b mod n a ≡ b mod Determine the highest multiple of $$2$$2 that is less than or equal to $$4$$4. Aug 30, 2009 · 167 Mod just means you take the remainder after performing the division. 1. More generally, the Python supports a wide range of arithmetic operators that you can use when working with numbers in your code. If you are asking “what is 4 mod 4?” then what you really need to know is “what is the remainder when I divide 4 by 4?”. Mod calculator gets two inputs to find mod with formula of modular arithmetic calculator. Modular arithmetic is a system of arithmetic for integers where numbers "wrap around" after reaching a certain value (the modulus). It is used in cryptography and computer science for efficient computation. What is the remainder of dividing 700. This is all I Calculate 2 modulo 4, also known as 2 mod 4. When you submit your Dividend and the Divisor below, we will show you two different ways of finding the Remainder. According to the Theorem, if in a given set of equations, each equation has a different number (say, n1, n2, , nk), and these numbers are all relatively prime Use the remainder calculator to find the quotient and remainder of division. There are lots of reasons why you would want to use modulo, including checking if a number is even or odd, counting something for a certain - It finds the remainder when one number is divided by another. Using the result of the last question, show that integers of the form 4t+3 May 22, 2025 · About Remainders and Modular Arithmetic When one integer is divided by another, the remainder is what's left over from the division operation. But -21 divided by 4 with truncation towards 0 (as C's / operator) gives -5 with a remainder (C -21 % 4) of -1. 2 mod 4. ) It should be clear now that a≡b(mod m) if and only if aand bhave the same remainders when divided by m. Modulo calculator finds the modulus when two numbers are being divided. 2. See all calculation steps. Modulo Calculator uses values of two real numbers, to find out the remainder after division of one number by another. The result of 7 modulo 5 is 2 because the remainder of 7 / 5 is 2. In this tutorial, you’ll learn: How modulo works in mathematics How to use the Python modulo operator with different numeric types How Python calculates the results of a modulo Step 2: Now click the button “Solve ” to get the remainder Step 3: Finally, the remainder and quotient will be displayed in the output field What is Meant by Remainder? In Mathematics, the division process is the opposite of the multiplication process. In your case, you are looking at 2 mod 4. For example, if you divide 17 by 5, the result is 3 with a remainder of 2 — so 17 mod 5 = 2. 7. For example: 1 4 m o d 5 = 4 14 mod 5 = 4 because 5 fits into 14 twice, with 4 left over. Examples: The result of 10 modulo 5 is 0 because the remainder of 10 / 5 is 0. Therefore, we can express this as: Modulo, often denoted as % in programming or represented as "mod," is a fundamental mathematical operation that calculates the remainder when one number is divided by another. When we say a ≡ b (mod n), we are saying that a and b are in the same remainder class modulo n. Interested in modulo operations? We are going to discuss modulo definition, how to find mod using a dividend and divisor, how to use modulus calculator, modulo arithmetic Dec 29, 2019 · When you see the % symbol, you may think "percent". 42^-1 mod 5 Definition and Usage The MOD () function returns the remainder of a number divided by another number. For example, in 17 m o d 5 17 mod 5, dividing 17 by 5 gives a quotient of 3 and a remainder of 2. Aug 1, 2025 · Modulo calculator finds a mod b, the remainder when a is divided by b. -I sort of know how to use the Euclidean Algorithm, but I'm not really sure what it has to do with mod. Adding 4 hours to 9 o'clock gives 1 o'clock, since 13 is congruent to 1 modulo 12. Time-keeping on this clock uses arithmetic modulo 12. Nov 26, 2020 · What does mod 4 mean? Put simply, modulo is the math operation of finding the remainder when you divide two numbers together. The glitch I am going to point out is not very serious, though keeping tracking of such things helps. Use our free Remainder Calculator to quickly find quotients and remainders in division problems. To find the remainder when a number is divided by 4, you can look at the last two digits of the number: Mar 12, 2025 · Perform modular arithmetic calculations online. FAQ What is a quotient? In arithmetic, a quotient is the quantity produced by the division of two numbers. The Mod operator in Excel VBA gives the remainder of a division. Modulo Calculator with steps computes mod between two numbers. Enter 2 numbers to find the remainder of their division. Is 7 ≡ 15(mod5) 7 ≡ 15 The modulo calculator finds the remainder of the division of two rational/irrational positive/negative numbers. In mathematics, modular arithmetic is a system of arithmetic operations for integers, other than the usual ones from elementary arithmetic, where numbers "wrap around" when reaching a certain value, called the modulus. 7 ≡ −3 (mod 5) follows since 7 − (−3) = 10 is divisible by 5. Why is 2 mod 4? - Because 2 divided by 4 leaves a remainder of 2. Simplify complex math effortlessly. So like 10 mod 2 is 0, cause 10 divides into 2 evenly without a remainder. All About Modulo Calculator: Some math ideas are about measuring how much. 5. Oct 11, 2023 · Modulo (Mod) Calculator - Your go-to tool for precise modulo operations on any number, big or small. See how it works by examples: floor (2. In mathematics, this operation is known as the "modulo" or "mod" function. 15 mod 4 = 3 Explanation: When 15 is divided by 4, the quotient is 3, and the remainder is 3. For instance, if you were dividing a by b and there was a remainder of n, then the result of a mod b would be n. Jan 11, 2024 · 1/2 mod 4: Here's how to calculate 1/2 modulo 4 step by step, including all terms and useful information. Hence, every ais congruent modulo m to one and only one of the integers in the set {0,1,2,,m−1}, namely the (unique) remainder when divided by m. The modulo operation returns the remainder in division of 2 positive or negative numbers or decimals. 8. Jul 23, 2025 · Chinese Remainder Theorem is a mathematical principle that solves systems of modular equations by finding a unique solution from the remainder of the division. Therefore 3 2022 has remainder 4 when divided by 5. Show that if n is an integer, then n2 ≡ 0(mod4) or n2 ≡1(mod4). Calculate seven hundred point two modulo four point eight. For example: -21 mod 4 is 3 because -21 + 4 x 6 is 3. Converting everyday terms to math, an “even number” is one where it’s “0 mod 2” — that is, it has a remainder of 0 when divided by 2. Explore More Math Tools a mod 1 is always 0; a mod 0 is undefined; Divisor (b) must be positive. 42 mod 5 2. 8 and 12 both leave a remainder of 0 when divided by 4, so they Modulo Operator as Used in our Calculator This operation (or function) rounds a value downwards to the nearest integer even if it is already negative. Quickly find remainders, understand calculations, and apply mod operations in real-life scenarios. You can also find out how to find modulus manually. Understanding Modulo Operation The modulo operation, often represented as a mod b, gives the remainder of the division of a by b. In this case, since 4 can't go into 2, the remainder is 2, so 2%4 is 2. How do you calculate modulo? - Divide the dividend by the divisor, then find the remainder. In fact, for every example in Preview Activity 7. Modulo is frequently expressed as a mod b; however, in some cases, it can be expressed as a % b. a) X = 1 (mod 5) X = 2 (mod 7) X = 3 (mod 9) X = 4 (mod 11 Find the remainder instantly with our free Remainder Calculator. 1 Chinese Remainder Theorem Using the techniques of the previous section, we have the necessary tools to solve congruences of the form ax b (mod n). This is often used by programmers and computer scientists. This is the method used in our calculator. klpwdda egq tttmrazf leysr bmcy ehrmcry fwqrycv vofp llwyyt djjyxa