Mastering Modular Arithmetic: A Guide to Addition, Subtraction, and Multiplication Rules

The rules for ordinary arithmetic involving addition, subtraction, and multiplication carry over into modular arithmetic.

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Modular arithmetic is a branch of arithmetic that deals with the remainder when an integer is divided by another integer called the modulus. The rules for ordinary arithmetic involving addition, subtraction, and multiplication work similarly in modular arithmetic, with some slight modifications to account for the nature of modular arithmetic.

Addition in modular arithmetic follows the same rules as regular addition. To add two numbers modulo n, add them as usual and then take the remainder when the sum is divided by n. For example, to perform 5 + 7 ≡ ?, mod 4, we add 5 and 7 to get 12 and then take the remainder when 12 is divided by 4, which is 0. Therefore, 5 + 7 ≡ 0, mod 4.

Subtraction in modular arithmetic also follows the same rules as regular subtraction. To subtract two numbers modulo n, subtract them as usual and then take the remainder when the difference is divided by n. For example, to perform 7 – 5 ≡ ?, mod 4, we subtract 5 from 7 to get 2 and then take the remainder when 2 is divided by 4, which is 2. Therefore, 7 – 5 ≡ 2, mod 4.

Multiplication in modular arithmetic also follows the same rules as regular multiplication. To multiply two numbers modulo n, multiply them as usual and then take the remainder when the product is divided by n. For example, to perform 5 × 7 ≡ ?, mod 4, we multiply 5 and 7 to get 35 and then take the remainder when 35 is divided by 4, which is 3. Therefore, 5 × 7 ≡ 3, mod 4.

However, there are some rules specific to modular arithmetic involving division and exponents. Division is not always defined in modular arithmetic and can only be performed under certain conditions. Exponents in modular arithmetic have to be computed using a method known as modular exponentiation.

In conclusion, the rules for ordinary arithmetic involving addition, subtraction, and multiplication carry over into modular arithmetic with some modifications to account for the nature of modular arithmetic.

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