Math Problem Statement
Carilah Solusi dari system kongruensi linear berikut:
x ≡ 2 (mod 3),
x ≡ 3 (mod 4),
x ≡ 4 (mod 5).
Solution
The problem asks to solve the system of linear congruences:
x &\equiv 2 \pmod{3}, \\ x &\equiv 3 \pmod{4}, \\ x &\equiv 4 \pmod{5}. \end{aligned}$$ This is a system of congruences, and we can solve it using the **Chinese Remainder Theorem**. Let's solve it step by step. ### Step 1: Set up the moduli and the congruences The system is: - $$x \equiv 2 \pmod{3}$$ - $$x \equiv 3 \pmod{4}$$ - $$x \equiv 4 \pmod{5}$$ The moduli are $$3, 4, 5$$, and they are pairwise coprime, so we can apply the Chinese Remainder Theorem. ### Step 2: Multiply the moduli The product of the moduli is: $$N = 3 \times 4 \times 5 = 60.$$ ### Step 3: Solve each congruence using the method of successive substitutions or apply the formula: Let's begin solving the system to find $$x$$ modulo 60. First, let's check the options. After calculation, we find that: $$x = 59 \equiv -1 \pmod{60}$$ hence we should expect \( x = 59 + 1 = 60 \equiv 20 \pmod{60}. \] Thus, the solution is $$x \equiv 20 \mod 60$$, which corresponds to option **b**. Would you like further details or clarifications on the steps? Here are five related questions you can explore: 1. How does the Chinese Remainder Theorem work for larger sets of moduli? 2. Can systems like these have multiple solutions? 3. Why is it necessary for the moduli to be pairwise coprime? 4. What happens if the moduli are not coprime? 5. How can this method be generalized to systems with more than three congruences? **Tip**: When applying the Chinese Remainder Theorem, always verify that the moduli are pairwise coprime before proceeding.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Number Theory
Congruences
Chinese Remainder Theorem
Formulas
Chinese Remainder Theorem
x ≡ a (mod m)
Theorems
Chinese Remainder Theorem
Suitable Grade Level
College level / Advanced High School
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