1. **State the problem:** Solve the congruence equation $$13x \equiv 71 \pmod{380}$$. 2. **Formula and rules:** To solve $$ax \equiv b \pmod{m}$$, we need to find the modular inverse of $$a$$ modulo $$m$$ if it exists. The modular inverse $$a^{-1}$$ satisfies $$a \cdot a^{-1} \equiv 1 \pmod{m}$$. 3. **Check gcd:** Compute $$\gcd(13, 380)$$ to ensure the inverse exists. $$\gcd(13, 380) = 1$$ since 13 is prime and does not divide 380. 4. **Find modular inverse of 13 mod 380:** Use the Extended Euclidean Algorithm. - Express $$1$$ as a linear combination of $$13$$ and $$380$$: $$380 = 13 \times 29 + 3$$ $$13 = 3 \times 4 + 1$$ $$1 = 13 - 3 \times 4$$ Substitute $$3 = 380 - 13 \times 29$$: $$1 = 13 - (380 - 13 \times 29) \times 4 = 13 - 380 \times 4 + 13 \times 29 \times 4 = 13 \times (1 + 116) - 380 \times 4 = 13 \times 117 - 380 \times 4$$ So, $$13 \times 117 \equiv 1 \pmod{380}$$ Thus, the modular inverse of $$13$$ modulo $$380$$ is $$117$$. 5. **Solve for $$x$$:** Multiply both sides of the original congruence by $$117$$: $$13x \equiv 71 \pmod{380}$$ $$\Rightarrow 117 \times 13x \equiv 117 \times 71 \pmod{380}$$ $$\Rightarrow (\cancel{117 \times 13}) x \equiv 117 \times 71 \pmod{380}$$ Since $$117 \times 13 \equiv 1 \pmod{380}$$, $$x \equiv 117 \times 71 \pmod{380}$$ Calculate $$117 \times 71$$: $$117 \times 71 = 8307$$ Reduce modulo 380: $$8307 \div 380 = 21$$ remainder $$327$$ So, $$x \equiv 327 \pmod{380}$$ 6. **Final answer:** The solution to the congruence is $$\boxed{x \equiv 327 \pmod{380}}$$.