Math Problem Statement
4x = 10 ( mod 14 )
Solution
The given equation is:
We need to solve this congruence equation for . Let's break it down step by step:
Step 1: Simplify the equation
Since this is a modulo equation, we're working in the system of integers modulo 14. The equation reads that is congruent to 10 modulo 14, which means:
Step 2: Check if a solution exists
We want to simplify . First, note that 4 and 14 have a greatest common divisor (GCD) of 2. Therefore, the equation can only have a solution if the right-hand side, 10, is divisible by this GCD (2), which it is.
Step 3: Divide by the GCD (which is 2)
We can now divide the entire equation by 2:
Step 4: Solve the new congruence
Now, we need to solve:
We need to find the multiplicative inverse of 2 modulo 7. To find this, we test values of such that:
Through trial or using the Extended Euclidean Algorithm, we find that , because:
Step 5: Multiply both sides by the inverse of 2 (which is 4)
Now, multiply both sides of by 4:
Since , we have:
Step 6: General solution
Now that we know , we can write the general solution. Since we divided by 2 earlier, the original modulus was 14, so the general solution is:
Thus, the solution to the equation is:
Would you like further details or have any questions?
Here are 5 related questions:
- How do we find the multiplicative inverse in modular arithmetic?
- What happens if the GCD is not a divisor of the right-hand side in a modular equation?
- Can you explain how the Extended Euclidean Algorithm helps in finding the inverse?
- How do you determine if a modular equation has no solution?
- What are some real-world applications of modular arithmetic?
Tip: In modular arithmetic, when solving , always check the GCD of and to see if a solution exists!
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Math Problem Analysis
Mathematical Concepts
Modular Arithmetic
Congruence Equations
Greatest Common Divisor (GCD)
Multiplicative Inverse
Formulas
ax ≡ b (mod m)
Multiplicative Inverse: a^(-1) (mod m) such that a * a^(-1) ≡ 1 (mod m)
Theorems
Extended Euclidean Algorithm
Solving Linear Congruences
Suitable Grade Level
Grades 9-12