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The goal of this exercise is to practice finding the inverse modulo m of some (relatively prime) integer n. We will find the inverse of 5 modulo 22, i.e., an integer c such that 5c = 1 (mod 22). First we perform the Euclidean algorithm on 5 and 22: 22 = 4* + *2 +1 [Note your answers on the second row should match the ones on the first row.] Thus gcd(5,22)=1, i.e., 5 and 22 are relatively prime. Now we run the Euclidean algorithm backwards to write 1 = 22s + 5t for suitable integers s, t. S= t= when we look at the equation 22s + 5t = 1 (mod 22), the multiple of 22 becomes zero and so we get 5t = 1 (mod 22). Hence the multiplicative inverse of 5 modulo 22 is

The goal of this exercise is to practice finding the inverse modulo m of some (relatively prime) integer n. We will find the inverse of 5 modulo 22, i.e., an integer c such that 5c = 1 (mod 22). First we perform the Euclidean algorithm on 5 and 22: 22 = 4* + *2 +1 [Note your answers on the second row should match the ones on the first row.] Thus gcd(5,22)=1, i.e., 5 and 22 are relatively prime. Now we run the Euclidean algorithm backwards to write 1 = 22s + 5t for suitable integers s, t. S= t= when we look at the equation 22s + 5t = 1 (mod 22), the multiple of 22 becomes zero and so we get 5t = 1 (mod 22). Hence the multiplicative inverse of 5 modulo 22 is

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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The goal of this exercise is to practice finding the inverse modulo m of some (relatively prime) integer n. We will find the inverse of 5 modulo 22, i.e., an integer c such that 5c = 1 (mod 22). First we perform the Euclidean algorithm on 5 and 22: 22 4* *2 +1 [Note your answers on the second row should match the ones on the first row.] = Thus gcd(5,22)=1, i.e., 5 and 22 are relatively prime. Now we run the Euclidean algorithm backwards to write 1 = 22s + 5t for suitable integers s, t. S = t= when we look at the equation 22s + 5t = 1 (mod 22), the multiple of 22 becomes zero and so we get 5t = 1 (mod 22). Hence the multiplicative inverse of 5 modulo 22 is
Transcribed Image Text:The goal of this exercise is to practice finding the inverse modulo m of some (relatively prime) integer n. We will find the inverse of 5 modulo 22, i.e., an integer c such that 5c = 1 (mod 22). First we perform the Euclidean algorithm on 5 and 22: 22 4* *2 +1 [Note your answers on the second row should match the ones on the first row.] = Thus gcd(5,22)=1, i.e., 5 and 22 are relatively prime. Now we run the Euclidean algorithm backwards to write 1 = 22s + 5t for suitable integers s, t. S = t= when we look at the equation 22s + 5t = 1 (mod 22), the multiple of 22 becomes zero and so we get 5t = 1 (mod 22). Hence the multiplicative inverse of 5 modulo 22 is
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