Math Problem Statement

Solve the linear congruence 74x ≡ 7 (mod 69).

Solution

The problem asks you to solve the linear congruence:

74x7(mod69)74x \equiv 7 \pmod{69}

Step-by-Step Solution:

  1. Simplify the equation mod 69: 74x7(mod69)74x \equiv 7 \pmod{69} Notice that 745(mod69)74 \equiv 5 \pmod{69}, since 7469=574 - 69 = 5. So the equation simplifies to: 5x7(mod69)5x \equiv 7 \pmod{69}

  2. Find the multiplicative inverse of 5 mod 69: To solve 5x7(mod69)5x \equiv 7 \pmod{69}, we need to find the inverse of 5 modulo 69. This can be done using the extended Euclidean algorithm.

    Applying the algorithm: 69=13×5+469 = 13 \times 5 + 4 5=1×4+15 = 1 \times 4 + 1 4=4×1+04 = 4 \times 1 + 0 Working backward, we find: 1=51×(6913×5)=14×51×691 = 5 - 1 \times (69 - 13 \times 5) = 14 \times 5 - 1 \times 69 So, the inverse of 5 mod 69 is 1414, since 14×51(mod69)14 \times 5 \equiv 1 \pmod{69}.

  3. Solve for x: Multiply both sides of the simplified congruence 5x7(mod69)5x \equiv 7 \pmod{69} by the inverse of 5, which is 14: x14×7(mod69)x \equiv 14 \times 7 \pmod{69} Calculate: 14×7=9814 \times 7 = 98 Now reduce 98 modulo 69: 9829(mod69)98 \equiv 29 \pmod{69}

So, the solution is: x29(mod69)x \equiv 29 \pmod{69}

Final Answer:

x=29(mod69)x = 29 \pmod{69}

Let me know if you need more details or explanations!

Here are 5 related questions to deepen your understanding:

  1. What is the general method to find the inverse of a number in modular arithmetic?
  2. How would the solution change if the modulus were different, say mod 70?
  3. Can all linear congruences be solved using the method outlined?
  4. What if the coefficient of xx and the modulus are not coprime? How do you handle that case?
  5. How does the extended Euclidean algorithm work, and why is it useful in modular arithmetic?

Tip: When solving linear congruences, always check if the greatest common divisor (GCD) of the coefficient of xx and the modulus is 1, as this ensures that an inverse exists.

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Math Problem Analysis

Mathematical Concepts

Linear Congruences
Modular Arithmetic
Extended Euclidean Algorithm

Formulas

74x ≡ 7 (mod 69)
Inverse of a number modulo n

Theorems

Modular Inverses
Extended Euclidean Algorithm

Suitable Grade Level

Grades 11-12