Mastering Linear Congruence: Step-by-Step Solutions to Commonly Seen Equations

🔍 In This Video:
- Background on Linear Congruence: Discover the foundational principles of linear congruence equations and why they're a cornerstone in modular arithmetic.
-Real-World Examples: Follow along as we solve a variety of linear congruence equations, from the simple to the complex, including cases where the greatest common divisor (GCD) plays a pivotal role.
- Euclidean Algorithm: Learn how to apply the Euclidean algorithm to find solutions efficiently.
- Practical Techniques: Gain practical strategies to tackle your own equations and enhance your problem-solving toolkit.
🛠️ Solved Equations:
- 256x ≡ 179 (mod 337)
- 20x ≡ 7 (mod 10) - No solution case
- 12x ≡ 16 (mod 20)
- 1296x ≡ 1125 (mod 1935) - Transition to a simpler equation
📚 Educational Value:
- Step-by-step walkthroughs of solutions.
- Clear explanations of complex concepts.
- Visual representations with pen-and-paper demonstrations.
00:00 Introduction to Linear Congruence Equations
00:33 Basic Theorem on Existence of Solutions
00:43 Special Case of (a, m) =1 Co-prime
01:07 Start of Problem Walk-Through
02:07 Solve 256x ≡ 179 (mod 337)
02:39 Walk-through of Euclidean Algorithm
06:34 Solution for 256x ≡ 179 (mod 337)
08:15 An Example of Non-existence Solution 20x ≡ 7 (mod 10)
09:00 Solve 12x ≡ 16 (mod 20)
11:19 Solve 1296x ≡ 1125 (mod 1935)
16:06 Review of the Steps
✅ By the end of this video, you will:
- Understand the concept of linear congruence and its applications.
- Be able to solve linear congruence equations using the Euclidean algorithm.
- Know how to handle equations with no solutions.
- Be equipped to find multiple solutions in more complex scenarios.
Remember to hit the 'Like' button if you find this video helpful and 'Subscribe' for more mathematical insights. Have questions or want to see more on this topic? Drop a comment below!
"Mastering Linear Congruence" is more than just a tutorial; it's your stepping stone to excelling in mathematics. Let's solve the puzzle of linear congruence equations together!
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[@ghita2003able]

thank you , very clear explanation !

[@amyjohnson5609]

thank you, You saved me for my exam

[@ullas.nbanglore6512]

If we get negative value of x how to make it positive

[@ompatel6849]

-106 * 125 = -13250 -13250 mod 215 = -135 to make it positive we add 215, to make it 80. Note that congruency is about cycles so we can add the mod to get to a positive solution.

[@ullas.nbanglore6512]

Thank you

[@physicistramatv4203]

congruence pls

[@physicistramatv4203]

solve for me this question ,23x^2-8x+34=0mod77

[@thinkinginmath3009]

I can provide a hint: use the Chinese Remainder Theorem (CRT). It involves solving the congruence modulo the prime factors of 77 (i.e., 7 and 11) and then combining the solutions.