To solve the equation 25x ≡ 15 (mod 29), we need to find the value of x that satisfies the congruence.
First, let’s find the modular inverse of 25 modulo 29. The modular inverse of a number a modulo m is another number b such that (a * b) ≡ 1 (mod m).
To find the modular inverse of 25 (mod 29), we can use the extended Euclidean algorithm.
Extended Euclidean Algorithm: 29 = 1 * 25 + 4 25 = 6 * 4 + 1
1 = 25 – 6 * 4 = 25 – 6 * (29 – 1 * 25) = 7 * 25 – 6 * 29
Therefore, the modular inverse of 25 modulo 29 is 7.
Now, to find the value of x, we multiply both sides of the congruence by the modular inverse (7):
7 * 25x ≡ 7 * 15 (mod 29)
175x ≡ 105 (mod 29)
Since we’re working modulo 29, we can reduce the coefficients:
8x ≡ 19 (mod 29)
To find the value of x, we can multiply both sides of the congruence by the modular inverse of 8 modulo 29, which is 11:
11 * 8x ≡ 11 * 19 (mod 29)
88x ≡ 209 (mod 29)
Reducing the coefficients:
2x ≡ 2 (mod 29)
Now, we can divide both sides by 2:
x ≡ 1 (mod 29)
Therefore, the solution to the congruence 25x ≡ 15 (mod 29) is x ≡ 1 (mod 29).