Answer :
The solution to 19x ≡ 4 (mod 141) using the modular inverse of 55 modulo 89 is x ≡ 16 (mod 141).
To find the inverse of 19 modulo 141 using the Euclidean algorithm, we can follow these steps:
1: Apply the Euclidean algorithm to find the greatest common divisor (gcd) of 19 and 141.
141 = 7 * 19 + 8
19 = 2 * 8 + 3
8 = 2 * 3 + 2
3 = 1 * 2 + 1
2: Rewriting each equation in terms of remainders:
8 = 141 - 7 * 19
3 = 19 - 2 * 8
2 = 8 - 2 * 3
1 = 3 - 1 * 2
3: Working backward, substitute the previous equations into the last equation to express 1 in terms of 19 and 141:
1 = 3 - 1 * 2
= 3 - 1 * (8 - 2 * 3)
= 3 * 3 - 1 * 8
= 3 * (19 - 2 * 8) - 1 * 8
= 3 * 19 - 7 * 8
= 3 * 19 - 7 * (141 - 7 * 19)
= 58 * 19 - 7 * 141
From the last equation, we can see that the Bézout coefficient of 19 is 58.
The last nonzero remainder in the Euclidean algorithm is 1.
Therefore, the inverse of 19 modulo 141 is 58.
To solve 19x = 4 (mod 141) using the modular inverse of 55 modulo 89, we can use the following steps:
1: Find the inverse of 55 modulo 89.
Apply the Euclidean algorithm:
89 = 1 * 55 + 34
55 = 1 * 34 + 21
34 = 1 * 21 + 13
21 = 1 * 13 + 8
13 = 1 * 8 + 5
8 = 1 * 5 + 3
5 = 1 * 3 + 2
3 = 1 * 2 + 1
Working backward:
1 = 3 - 1 * 2
= 3 - 1 * (5 - 1 * 3)
= 2 * 3 - 1 * 5
= 2 * (8 - 1 * 5) - 1 * 5
= 2 * 8 - 3 * 5
= 2 * 8 - 3 * (13 - 1 * 8)
= 5 * 8 - 3 * 13
= 5 * (21 - 1 * 13) - 3 * 13
= 5 * 21 - 8 * 13
= 5 * 21 - 8 * (34 - 1 * 21)
= 13 * 21 - 8 * 34
= 13 * (55 - 1 * 34) - 8 * 34
= 13 * 55 - 21 * 34
= 13 * 55 - 21 * (89 - 1 * 55)
= 34 * 55 - 21 * 89
So, the inverse of 55 modulo 89 is 34.
2: Multiply both sides of the equation by the inverse of 55 modulo 89.
19x ≡ 4 (mod 141)
34 * 19x ≡ 34 * 4 (mod 141)
646x ≡ 136 (mod 141)
3: Reduce the coefficients and values modulo 141.
646x ≡ 136 (mod 141)
4x ≡ 136 (mod 141)
4: Solve for x.
To solve this congruence, we can multiply both sides by the inverse of 4 modulo 141, which is 71 (since 4 * 71 ≡ 1 (mod 141)):
71 * 4x ≡ 71 * 136 (mod 141)
284x ≡ 964 (mod 141)
Reducing coefficients modulo 141:
2x ≡ 32 (mod 141)
Now, we can solve this congruence to find x:
x ≡ 16 (mod 141)
Therefore, the solution to 19x ≡ 4 (mod 141) using the modular inverse of 55 modulo 89 is x ≡ 16 (mod 141).
Learn more about Euclidean algorithm from this link:
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