Modular Equation Solving | x2 + 8x ≡ 0 (mod 56) | Step-by-Step Guide

  • Thread starter Cheesycheese213
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Notice that 56 = 7 * 8, so you can use the Chinese Remainder Theorem and solve the equation mod 7 and mod 8 separately, and then combine the solutions using the CRT. In summary, the modular equation x2 + 8x ≡ 0 (mod 56) can be solved by using the Chinese Remainder Theorem to solve the equation mod 7 and mod 8 separately, and then combining the solutions to find the final solution.
  • #1
Cheesycheese213
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Homework Statement


Solve the modular equation x2 + 8x ≡ 0 (mod 56)

Homework Equations


Let a, b = integer
n|(a - b), written as a b (mod n)

The Attempt at a Solution


I tried to separate them, and got
2|x2 + 8x or x2 + 8x ≡ 0 (mod 2) as well as
7|x2 + 8x or x2 + 8x ≡ 0 (mod 7)
but I'm a bit stuck now, and I'm pretty sure they might be wrong?

How should I be solving this? Thanks!
 
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  • #2
Cheesycheese213 said:

Homework Statement


Solve the modular equation x2 + 8x ≡ 0 (mod 56)

Homework Equations


Let a, b = integer
n|(a - b), written as a b (mod n)

The Attempt at a Solution


I tried to separate them, and got
2|x2 + 8x or x2 + 8x ≡ 0 (mod 2) as well as
7|x2 + 8x or x2 + 8x ≡ 0 (mod 7)
but I'm a bit stuck now, and I'm pretty sure they might be wrong?

How should I be solving this? Thanks!
Can't you write the original equation as ##x(x + 8) \equiv 0 \mod 56##?
 

Related to Modular Equation Solving | x2 + 8x ≡ 0 (mod 56) | Step-by-Step Guide

1. What is a modular equation?

A modular equation is an equation that involves the concept of congruence, where the solution is restricted to a certain set of numbers. In this case, the equation x2 + 8x ≡ 0 (mod 56) means that the remainder when dividing x2 + 8x by 56 is equal to 0.

2. How do you solve a modular equation?

To solve a modular equation such as x2 + 8x ≡ 0 (mod 56), you need to find the values of x that satisfy the equation. This can be done by first simplifying the equation, then finding the inverse of the coefficient of x (if it exists). The solutions will then be the numbers that satisfy the equation when substituted for x.

3. What is the purpose of solving a modular equation?

Solving a modular equation can help in solving problems related to congruence, such as finding the remainder when dividing large numbers or determining if two numbers are relatively prime. It is also used in many branches of mathematics, including cryptography and number theory.

4. What are the steps for solving the equation x2 + 8x ≡ 0 (mod 56)?

Step 1: Simplify the equation by factoring out the common factor of x.

Step 2: Rewrite the equation as x(x + 8) ≡ 0 (mod 56).

Step 3: Find the inverse of 56 (mod 56), which is 1.

Step 4: Multiply both sides of the equation by the inverse to eliminate the modulus.

Step 5: Solve for x by finding the values that satisfy the equation.

5. Can you provide a step-by-step guide for solving the equation x2 + 8x ≡ 0 (mod 56)?

Step 1: Simplify the equation by factoring out the common factor of x.

x(x + 8) ≡ 0 (mod 56)

Step 2: Rewrite the equation as x(x + 8) ≡ 0 (mod 56).

Step 3: Find the inverse of 56 (mod 56), which is 1.

Step 4: Multiply both sides of the equation by the inverse to eliminate the modulus.

1 * x(x + 8) ≡ 0 * 1 (mod 56)

x(x + 8) ≡ 0 (mod 56)

Step 5: Solve for x by finding the values that satisfy the equation.

Since x(x + 8) ≡ 0 (mod 56), this means that either x ≡ 0 (mod 56) or (x + 8) ≡ 0 (mod 56).

Solving for x, we get x ≡ 0 (mod 56) or x ≡ 48 (mod 56).

Therefore, the solutions to the equation x2 + 8x ≡ 0 (mod 56) are x ≡ 0 (mod 56) or x ≡ 48 (mod 56).

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