Answer :
Final answer:
The inverse of 19 mod 141 is the number which, when multiplied with 19, results in 1 modulo 141. Using the Extended Euclidean Algorithm reveals that this inverse is 52.
Explanation:
The inverse of 19 mod 141 is the number which, when multiplied by 19, yields a product of 1 modulo 141. This is also commonly referred to as the modular multiplicative inverse. To compute this, you may use the Extended Euclidean Algorithm or lookup tables for small numbers.
To find the inverse using a systematic approach:
Apply the Extended Euclidean Algorithm to the pair (19, 141) to find the greatest common divisor (GCD) and the coefficients of Bézout's identity, which are the integers x and y that satisfy 19x + 141y = gcd(19, 141).
Since gcd(19, 141) should be 1 for the inverse to exist (19 and 141 must be coprime), the coefficient x in the identity, where 19x mod 141 = 1, will be the inverse of 19.
Using this method, you would find that the inverse of 19 mod 141 is 52, as 19 * 52 mod 141 equals 1.
Answer:
52
Step-by-step explanation:
Let a = 19 and m = 141
Step 1: Find the gcd(141, 19).
We do this using the Euclidean Algorithm
This takes advantage of the fact that if
a = bq + r for a, b, q, r ∈ Z, then gcd(a, b) = gcd(b, r).
Using this algorithm we get:
141 = 7 · 19 + 8 gcd(141, 19) = gcd(19, 8)
19 = 2 · 8 + 3 = gcd(8, 3)
8 = 2 · 3 + 2 = gcd(3, 2)
3 = 1 · 2 + 1 = gcd(2, 1)
2 = 2 · 1 + 0 = 1
Therefore, 141 and 19 are relatively prime and 19 mod 141 has an inverse.
The modular inverse of 19 mod 141 is 52.
