What is meant by modular arithmetic?
In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers “wrap around” when reaching a certain value, called the modulus. A familiar use of modular arithmetic is in the 12-hour clock, in which the day is divided into two 12-hour periods.
How do you do modular arithmetic?
The modulus is another name for the remainder after division. For example, 17 mod 5 = 2, since if we divide 17 by 5, we get 3 with remainder 2. Modular arithmetic is sometimes called clock arithmetic, since analog clocks wrap around times past 12, meaning they work on a modulus of 12.
Is modular arithmetic used in computer science?
Modular arithmetic also underlies public key cryptography systems, which are vital for modern commerce. It is also widely used in computer science.
What are the properties of modular arithmetic?
Properties of addition in modular arithmetic:
- If a + b = c a+b = c a+b=c, then a ( m o d N ) + b ( m o d N ) ≡ c ( m o d N ) .
- If a ≡ b ( m o d N ) a\equiv b\pmod N a≡b(modN), then a + k ≡ b + k ( m o d N ) a+k \equiv b+k \pmod N a+k≡b+k(modN) for any integer k .
How do you reduce modular arithmetic?
In modular arithmetic, when we say “reduced modulo ,” we mean whatever result we obtain, we divide it by n, and report only the smallest possible nonnegative residue. The next theorem is fundamental to modular arithmetic. Let n≥2 be a fixed integer. If a≡b (mod n) and c≡d (mod n), then a+c≡b+d(modn),ac≡bd(modn).
How do you multiply modular arithmetic?
You just multiply the two numbers and then calculate the standard name. For example, say the modulus is 7. Let’s look at some mod 15 examples. One thing to notice is that in modular arithmetic you can multiply two numbers that are both nonzero, and the result can be zero.
Can you divide in modular arithmetic?
Can we always do modular division? The answer is “NO”. In modular arithmetic, not only 4/0 is not allowed, but 4/12 under modulo 6 is also not allowed. The reason is, 12 is congruent to 0 when modulus is 6.
What is the difference between modular arithmetic and ordinary arithmetic?
The only difference between modular arithmetic and the arithmetic you learned in your primary school is that in modular arithmetic all operations are performed regarding a positive integer, i.e. the modulus. This establishes a natural congruence relation on the integers.
What are the quotient and remainder when − 11 is divided by 3?
Solution: The quotient when −11 is divided by 3 is −4 = −11 div3, and the remainder is 1 = −11 mod 3.
What is the formula of division algorithm?
The division algorithm states that for any integer, a, and any positive integer, b, there exists unique integers q and r such that a = bq + r (where r is greater than or equal to 0 and less than b). We call a the dividend, b the divisor, q the quotient, and r the remainder.
What is Algorithm Theorem?
Theorem. [ Division Algorithm] Suppose a > 0 and b are integers. Then there is a unique. pair of integers q and r such that. b = aq + r where 0 ≤ r < a.
What are the three parts of division?
Each part of a division equation has a name. The three main names are the dividend, the divisor, and the quotient. There are three special cases to consider when dividing. 1) Dividing by 1: When dividing something by 1, the answer is the original number.
What is divisibility theorem?
Divisibility is one of the most fundamental concepts in number theory. If a divides b, we say “a is a divisor of b”, “b is divisible by a”, “a is a factor of b”, “b is a multiple of a”. • Note: The “divisor” a in this definition can be negative, but must be nonzero; divisibility by 0 is not defined.
How do you do divisibility rules?
The Divisibility Rules
- Any integer (not a fraction) is divisible by 1.
- The last digit is even (0,2,4,6,8)
- The sum of the digits is divisible by 3.
- The last 2 digits are divisible by 4.
- The last digit is 0 or 5.
- Is even and is divisible by 3 (it passes both the 2 rule and 3 rule above)
What is the divisibility rule of 7?
Divisibility by 7. To check if a number is evenly divisible by 7: Take the last digit of the number, double it Then subtract the result from the rest of the number If the resulting number is evenly divisible by 7, so is the original number.
What are the properties of divisibility?
Properties of Divisibility
- The following properties are: (i) When a number is divisible by another number, it is also divisible by the factors of the number.
- For example, 48 is divisible by 12.
- For example, 12 is divisible by 2 and 3 as,
- For example, 6 is a factor of 18 since 18 ÷ 6 = 3.
- For example, 4 is a factor of 12 since, 12 = 4 × 3.
What is meant by division algorithm?
A division algorithm is an algorithm which, given two integers N and D, computes their quotient and/or remainder, the result of Euclidean division. Some are applied by hand, while others are employed by digital circuit designs and software. Variants of these algorithms allow using fast multiplication algorithms.
What is a divisibility rule for 2?
Divisibility Rule of 2 If a number is even or a number whose last digit is an even number i.e. 2,4,6,8 including 0, it is always completely divisible by 2. Example: 508 is an even number and is divisible by 2 but 509 is not an even number, hence it is not divisible by 2.
What is the divisibility rule of 11?
Here an easy way to test for divisibility by 11. Take the alternating sum of the digits in the number, read from left to right. If that is divisible by 11, so is the original number. So, for instance, 2728 has alternating sum of digits 2 – 7 + 2 – 8 = -11. Since -11 is divisible by 11, so is 2728.
What are the 11 factors?
Table of Factors and Multiples
| Factors | Multiples | |
|---|---|---|
| 1, 3, 9 | 9 | 36 |
| 1, 2, 5, 10 | 10 | 40 |
| 1, 11 | 11 | 44 |
| 1, 2, 3, 4, 6, 12 | 12 | 48 |
What is the divisibility rule of 7 and 11?
Rule: Take the last digit and cross it out from the original number. Then double it. Subtract it from the “new” number which is the original number excluding the last digit. If the difference is divisible by 7, then the original number must also be divisible by 7.
Is 1111111 is divisible by 11?
Consider the number 1111111. Since this number (1) is not divisible by 11, 1111111 is also not divisible by 11.
Is 1011011 is divisible by 11?
Answer: No, 1011011 is not divisible by 11.
Which of the following numbers is divisible by 11 3333333 1111111 22222222 none of these?
Consider the number 1111111. Since this number (1) is not divisible by 11, 1111111 is also not divisible by 11. Consider the number /b>. Since this number (0) is divisible by 11, /b> is also divisible by 11.
Which of the following is a prime number?
Prime Numbers : Example Question #3 If is the first number in a series of prime numbers, which of the following could not be the value of the last number in the series? Explanation: The primes, in order, are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53.