Solving Modular Arithmetic Problems: How to Explained

[XodoX]

How do I solve those problems?

Like,

Find some x such that x$$\equiv$$8 mod (18)

Find the inverse of 12 modulo 41

Solve 2x=7 mod (13)

I know it's easy, but I don't get it.

Let a and be be integers, and let m be a positive integer. Then a $$\equiv$$ b ( mod m) if and only if a mod m = b mod m

That's the explanation in the book. I'm not getting it. Can somebody please explain this modular arithmetic to me?

 

[XodoX]

Nobody knows? Maybe wrong forum.

 

[BWV]

mod arithmetic is just a system of integers that "starts" over at the mod #. Think about the 12 notes on a piano - c,c#,d,d# etc. once you get to b or 12 it just starts over. so 7 mod 13 could be (assuming you start at 1) 7, 20, 33, 46, etc

 

[bapowell]

Or hours on the clock -- they start over after 12. Compare military time with clock time: 20:00 hours is 8 o'clock because the clock time arithmetic is modular:

8 = 20 mod(12)

 

[blue_raver22]

Modular arithmetic is a mathematical tool that is used to solve problems involving remainders. It is particularly useful in number theory, cryptography, and computer science. To solve modular arithmetic problems, you need to understand the concept of congruence and the rules that govern it.

Congruence is a relationship between two numbers that have the same remainder when divided by a given number. In other words, if two numbers, a and b, have the same remainder when divided by m, then we say that a and b are congruent modulo m, denoted as a \equiv b ( mod m). For example, 8 \equiv 26 ( mod 6) because both 8 and 26 have a remainder of 2 when divided by 6.

To solve problems involving modular arithmetic, you can use the following steps:

1. Understand the question: Make sure you understand what the problem is asking for. This will help you determine what information you need to find and what operations you need to perform.

2. Use the definition of congruence: As stated earlier, a \equiv b ( mod m) if and only if a mod m = b mod m. This means that to find the solution to a modular arithmetic problem, you need to find the value of a mod m or b mod m.

3. Apply the rules of congruence: There are several rules that govern congruence, such as addition, subtraction, multiplication, and division. These rules state that if a \equiv b ( mod m), then a + c \equiv b + c ( mod m), a - c \equiv b - c ( mod m), a \times c \equiv b \times c ( mod m), and a/c \equiv b/c ( mod m). These rules can be used to simplify the problem and find the solution.

Now, let's apply these steps to the problems you mentioned:

1. Find some x such that x\equiv8 mod (18): This question is asking for a number that has a remainder of 8 when divided by 18. Using the definition of congruence, we can say that x mod 18 = 8. To find the solution, we can look for numbers that have a remainder of 8 when divided by 18, such as 26, 44, 62, etc. So, x could be any of these numbers.

2. Find the inverse of 12 modulo