Question

For this question you may assume that gcd(2345,2023)=7 and that 7=44×2345−51×2023. Find all solutions, if any, to the linear congruence 2023x+123≡242(mod2345). If you find that solutions do exist then you express them modulo 2345 .

Answer 1

To solve the linear congruence 2023x + 123 ≡ 242 (mod 2345), we can rewrite the equation as 2023x ≡ 119 (mod 2345). We then find the multiplicative inverse of 289 modulo 335 to solve for x, which is x ≡ 107 (mod 335). Finally, we express the solution in terms of modulo 2345 as x ≡ 107 (mod 2345).

Step-by-step explanation:

To solve the linear congruence 2023x + 123 ≡ 242 (mod 2345), we can rewrite the equation as 2023x ≡ 242 - 123 (mod 2345). Simplifying the right side, we have 2023x ≡ 119 (mod 2345). Since gcd(2345, 2023) = 7, the congruence has solutions if and only if 119 is divisible by 7. In this case, 119 is divisible by 7, so we can divide both sides of the congruence by 7 to get 289x ≡ 17 (mod 335).

Next, we need to find the multiplicative inverse of 289 modulo 335. The extended Euclidean algorithm can be used to find this inverse. By applying the Euclidean algorithm several times, we find that 289(198) + 335(-17) = 1. Therefore, 198 is the multiplicative inverse of 289 modulo 335.

Multiplying both sides of the congruence by 198, we get x ≡ 198(17) ≡ 3366 ≡ 107 (mod 335). Now we need to express the solution in terms of modulo 2345. Since 335 is a factor of 2345, we can express the solution as x ≡ 107 (mod 2345).