2 mod 4 remainder. Number Theory 4. 2. 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. 27^400 mod 619 6. The % symbol in Python is called the Modulo Operator. So if it is Sunday today, 100 days from now it will be Tuesday! Enter any number . This means they have the same remainder when divided by n. [1] Sunzi's original formulation Chinese remainder theorem. 7 mod 3 = 1 Explanation: When 7 is divided by 3, the quotient is 2, and the remainder is 1. Use our free Remainder Calculator to quickly find quotients and remainders in division problems. *floor(a. Perfect for students, teachers, and anyone needing fast, accurate division results. 25 0. Then a ≡ b (mod n) iff a and b leave the same remainder when divided by n. Souare each of these and reduce modulo 4 to see what you get. Hint: n is in one of the 4 remainder classes modulo 4 . In this case, 4 goes into 3 zero times with a remainder of 3. For example, “5 mod 3 = 2” which means 2 is the remainder when you divide 5 by 3. See step-by-step solutions, formulas, and examples to master division in maths. Is 7 ≡ 15(mod5) 7 ≡ 15 The modulo calculator finds the remainder of the division of two rational/irrational positive/negative numbers. 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. 19^24 mod 21 4. Aug 1, 2025 · Modulo calculator finds a mod b, the remainder when a is divided by b. 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. This is often used by programmers and computer scientists. When we say a ≡ b (mod n), we are saying that a and b are in the same remainder class modulo n. In your case, you are looking at 2 mod 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. Here we will explain what 4 mod 2 means and show how to calculate it. Find the Quotient: The quotient is 0 because 4 goes into 2 zero times. Modulo is the name given to the remainder value. 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?". For example, if you divide 17 by 5, the result is 3 with a remainder of 2 — so 17 mod 5 = 2. That is to say when 6 6 and 11 11 are divided by 5 5 the remainder is 1. 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. 2 mod 4. Notice that 3 4 = 81 = 1 in mod 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 modulo operation helps to determine the remainder. Oct 30, 2023 · The question 'How many is 2%4' is a **modulus **operation in mathematics. 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. Example Consider we take modulo 4. How to calculate modulus without a calculator? - Use long division to find the remainder. 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). 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. It is used in cryptography and computer science for efficient computation. Therefore 3 2022 has remainder 4 when divided by 5. Calculate seven hundred point two modulo four point eight. Since 4 goes into 2 zero times, you end up with a remainder of 2. Consider a number of the form $4q+2$, you'll have an issue. 75. 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. 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. Section 4. The mod function follows the convention that mod(a,0) returns a. The modulo operation calculates the remainder when a number is divided by another. 2 revisited). 4 7 Using Fermat's Little Theorem, which states for a prime and a base not divisible by 2. This is all I Calculate 2 modulo 4, also known as 2 mod 4. Here we will explain what 2 mod 4 means and show how to calculate it. 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. In this case, since 4 can't go into 2, the remainder is 2, so 2%4 is 2. 4 mod 2 is 0. Use our free Modulo Calculator to find modulus and remainder with step-by-step solutions. 2. Jul 15, 2020 · It follows the Euclidean division rule, meaning the remainder always has the same sign as the divisor. Modulo 1. FAQ What is a quotient? In arithmetic, a quotient is the quantity produced by the division of two numbers. Modulo calculator finds the modulus when two numbers are being divided. 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. It’s written as: a m o d n a mod n. By simple algebra, if 5k + 13 5 k + 13 is divided by 4 4, then if k ≤ −6 k ≤ 6 the remainder is 0. Jan 12, 2014 · The Chinese Remainder Theorem is a method to solve the following puzzle, posed by Sun Zi around the 4th Century AD. 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. Experience swift and accurate results every time! Modulo Calculator A quick and easy calculator to solve modulo operations. Multiply: Multiply the quotient by 4: 0 * 4 = 0. Try now! Jun 1, 2025 · The modulus operation finds the remainder after dividing one number (the dividend) by another (the divisor). The Chinese remainder theorem is a theorem which gives a unique solution to simultaneous linear congruences with coprime moduli. Adding 4 hours to 9 o'clock gives 1 o'clock, since 13 is congruent to 1 modulo 12. 8. 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). Modulo Calculator Welcome to the Modulo Calculator also known as the Mod Calculator or the Modulus Calculator. 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. But, however, what follows this error, $\bar a^2=\bar r^2$ is correct. 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. Perfect for time, scheduling, and programming tasks. 4 mod 2 is short for 4 modulo 2 and it can also be called 4 modulus 2. See all calculation steps. When you submit your Dividend and the Divisor below, we will show you two different ways of finding the Remainder. What is 4 mod 2? The mod or modulus or modulo is the remainder after dividing one number by another. 2, we saw that n and s. Explore More Math Tools a mod 1 is always 0; a mod 0 is undefined; Divisor (b) must be positive. Example: 14 mod 12 equals 2 Because 14 12 = 1 with a remainder of 2 So 14 o'clock becomes 2 o'clock. 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). b = mod(a,m) returns the remainder after division of a by m, where a is the dividend and m is the divisor. The modular calculator only takes dividend and divisor from the user to calculate the remainder after the division. 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. Quickly find remainders, understand calculations, and apply mod operations in real-life scenarios. Apply the same and find X for the given set of congruent equations using CRT. 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. Understanding Modulo Operation The modulo operation, often represented as a mod b, gives the remainder of the division of a by b. Example (3. The Mod operator in Excel VBA gives the remainder of a division. 75 0. Example 3. 42^-1 mod 5 Use our Modulo Calculator to find the remainder when dividing two numbers. 2 7. The two main things which we need for division problems are dividend and divisor. 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. 21 3 mod 4 is the remainder when 3 is divided by 4. Subtract this highest multiple from the dividend $$4$$4. 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). 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. Modulo is the operation of finding the Remainder when you divide two numbers. 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. One of these operators is the modulo operator (%), which returns the remainder of dividing two numbers. 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". In other words, n≡0(mod4), or n≡1(mod4), or n≡ 2(mod4), or n≡3(mod4). 42^-1 mod 5 Definition and Usage The MOD () function returns the remainder of a number divided by another number. Syntax of Modulus Operator If x and y are integers, then the expression: x % y; pronounced as "x mod y". You’ve seen it in real life, even if you didn’t know it. Modulo Calculator uses values of two real numbers, to find out the remainder after division of one number by another. Time-keeping on this clock uses arithmetic modulo 12. This page starts with some simple examples. 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. 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. 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. These 4 remainder classes are: [0]4,[1]4,[2]4,[3]4. Home Modulo Calculator Modulo Calculator Easily find the remainder when dividing two numbers with our modulo calculator. 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. Jul 10, 2023 · (a) To find the remainders when and 4 are divided by 7, we will use modular arithmetic. The reason your calculator says 113 modulo 120 = 113 is because 113 < 120, so it isn't doing any division. A modulo calculator is an online tool that helps you find the remainder when one number (the dividend) is divided by another (the divisor). For example: 1 4 m o d 5 = 4 14 mod 5 = 4 because 5 fits into 14 twice, with 4 left over. 2); // returns Use this tool to find the remainder when a number (the dividend) is divided by another number (the divisor). 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. In fact, for every example in Preview Activity 7. The remainder when $$4$$4 is divided by $$2$$2 is $$0$$0. Learn how to use it with applications. This is the method used in our calculator. The modulo operation, symbolized as "mod", finds the remainder of division between two numbers. The modulo division operator produces the remainder of an integer division which is also called the modulus of the operation. Mod calculator gets two inputs to find mod with formula of modular arithmetic calculator. 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. Thus the remainder is 7. This mathematical operation is abbreviated as ‘mod’. We perform the calculation: $$4 - 4 = 0$$4−4 = 0 Conclude the result. For example: -21 mod 4 is 3 because -21 + 4 x 6 is 3. The result of 7 modulo 5 is 2 because the remainder of 7 / 5 is 2. Calculation To calculate 2 mod 4, follow these steps: Divide: Divide 2 by 4. So x = 3 2022 = 3 4*505+2 = (3 4 ) 505 * 3 2 = 1 505 * 9 = 9 = 4 in mod 5. The Chinese remainder theorem calculator is here to find the solution to a set of remainder equations (also called congruences). This function is often called the modulo operation, which can be expressed as b = a - m. 4 3. 42 mod 5 2. For example, if you want to know 17 mod 5, the calculator instantly shows that 17 divided by 5 leaves a remainder of 2. 1); // returns number 2 floor (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. For positive values, there is no difference between Euclidean and truncating division. The % sign denotes a modulus (remainder) operation. Modulo is a way of finding the remainder when one number is divided by another. 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. In case the advantage isn’t clear, we recall our earlier example. But -21 divided by 4 with truncation towards 0 (as C's / operator) gives -5 with a remainder (C -21 % 4) of -1. 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). 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. 3^302 mod 5 3. In mathematics, this operation is known as the "modulo" or "mod" function. 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 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. 2 over 4. 2 mod 4 is short for 2 modulo 4 and it can also be called 2 modulus 4. Understanding the Modulo Operation and Remainder Classes The modulo operation is written as "mod," this is important to remainder classes. So like 10 mod 2 is 0, cause 10 divides into 2 evenly without a remainder. 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. 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 highest multiple is $$4$$4 itself. All About Modulo Calculator: Some math ideas are about measuring how much. This function is used in mathematics, where the result of the modulo operation is the remainder of the Euclidean division. Modulo is about measuring what’s left. Our next goal is to define an arithmetic with remainders, again without calculating quotients. Show that if n is an integer, then n2 ≡ 0(mod4) or n2 ≡1(mod4). 8. In particular, every a is congruent to its remainder when divided by n, and no two distinct remainders are congruent modulo n. The Chinese Remainder Theorem gives us a tool to consider multiple such congruences simultaneously. To find the modulus of a mod b, use mod calculator. 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. On calculators, modulo is often calculated using the mod () function: mod (a, b) = r. It is used in finding even/odd numbers, cyclic patterns, and leap year calculations. a ≡ b mod n a ≡ b mod Determine the highest multiple of $$2$$2 that is less than or equal to $$4$$4. Simplify complex math effortlessly. For instance, "5 mod 3" evaluates to 2, as 5 divided by 3 leaves a remainder of 2. You can also find out how to find modulus manually. Aug 30, 2009 · 167 Mod just means you take the remainder after performing the division. . Modulo is frequently expressed as a mod b; however, in some cases, it can be expressed as a % b. 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. The floor function returns the remainder with the same sign as the divisor. 2 mod 4 with+steps. 7 ≡ −3 (mod 5) follows since 7 − (−3) = 10 is divisible by 5. 5. 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. /m). 15 mod 4 = 3 Explanation: When 15 is divided by 4, the quotient is 3, and the remainder is 3. For instance, in (a), how is $0\leq r^2<4$. 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. What is the remainder of dividing 700. 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. 2); // returns Equivalent to 3 2022 = x in mod 5. 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. ) It should be clear now that a≡b(mod m) if and only if aand bhave the same remainders when divided by m. More generally, the Python supports a wide range of arithmetic operators that you can use when working with numbers in your code. (Hence the justification of Gauss’ phrasearithmetic of remainders. How do you calculate modulo? - Divide the dividend by the divisor, then find the remainder. 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. 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. 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. 7^106 mod 143 5. Examples: The result of 10 modulo 5 is 0 because the remainder of 10 / 5 is 0. See how it works by examples: floor (2. Enter 2 numbers to find the remainder of their division. 25, and if k k is greater than −6 6 the remainder is 0. n/ were congruent modulo 9 since they both had the same remainder when divided by 9. 1. 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. Master modular arithmetic with our power mod calculator, perfect for calculations with exponents. 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. Jul 12, 2025 · In C or C++, the modulo operator (also known as the modulus operator), denoted by %, is an arithmetic operator. 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. 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. 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). -I sort of know how to use the Euclidean Algorithm, but I'm not really sure what it has to do with mod. 7); // returns number 2 floor (-5. Modular arithmetic can be used in addition, subtraction, multiplication, and exponentiation. Oct 11, 2023 · Modulo (Mod) Calculator - Your go-to tool for precise modulo operations on any number, big or small. 4: Find the remainder when 71453 7 1453 is divided by 8. But in Python, as well as most other programming languages, it means something different. How do you calculate mod 4? 1 mod 4 equals 1, since 1/4 = 0 with a remainder of 1. 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 glitch I am going to point out is not very serious, though keeping tracking of such things helps. Subtract: Subtract this Feb 23, 2022 · One value is the quotient, and the other is the remainder. 4. 7. Modular arithmetic is a system of arithmetic for integers where numbers "wrap around" after reaching a certain value (the modulus). 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. The modulo operation returns the remainder in division of 2 positive or negative numbers or decimals. Why is 2 mod 4? - Because 2 divided by 4 leaves a remainder of 2. Download the practice workbook, modify data, find new results and exercise! Find out what is 700. This is useful in lots of different situations, and especially in programming. 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?”. 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. There is no need for us to express 7 and −3 using the division algorithm. Modulo Calculator with steps computes mod between two numbers. zhyrhha vak avacdzw vmqugv xmge gui ynll eshtif exln jvfjqe