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4. Letn > 1 be an integer and a an integer with 0 < a < n. Then an integer s is an inverse modulo n of a if as = 1 (mod n). a. Prove that if ged(a, n) = 1, then a has an inverse modulo n. Begin by using the Extended Euclidean Algorithm to write gcd(a, n) as a linear combination of a and n 1 = gcd(a, n) = ax + ny for some integers x, y. Then apply "mod n" to both sides of the equation to show that x mod n is the inverse of a modulo n. b. Use the method from part (a) to compute the inverse modulo 15 of 7.

4. Letn > 1 be an integer and a an integer with 0 < a < n. Then an integer s is an inverse modulo n of a if as = 1 (mod n). a. Prove that if ged(a, n) = 1, then a has an inverse modulo n. Begin by using the Extended Euclidean Algorithm to write gcd(a, n) as a linear combination of a and n 1 = gcd(a, n) = ax + ny for some integers x, y. Then apply "mod n" to both sides of the equation to show that x mod n is the inverse of a modulo n. b. Use the method from part (a) to compute the inverse modulo 15 of 7.

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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4. Letn> 1 be an integer and a an integer with 0 < a <n. Then an integer s is an inverse modulo n of a if as = 1 (mod n). a. Prove that if gcd(a, n) = 1, then a has an inverse modulo n. Begin by using the Extended Euclidean Algorithm to write ged(a, n) as a linear combination of a and n 1 = gcd(a, n) = ax + ny for some integers x, y. Then apply "mod n" to both sides of the equation to show that x mod n is the inverse of a modulo n. b. Use the method from part (a) to compute the inverse modulo 15 of 7.
Transcribed Image Text:4. Letn> 1 be an integer and a an integer with 0 < a <n. Then an integer s is an inverse modulo n of a if as = 1 (mod n). a. Prove that if gcd(a, n) = 1, then a has an inverse modulo n. Begin by using the Extended Euclidean Algorithm to write ged(a, n) as a linear combination of a and n 1 = gcd(a, n) = ax + ny for some integers x, y. Then apply "mod n" to both sides of the equation to show that x mod n is the inverse of a modulo n. b. Use the method from part (a) to compute the inverse modulo 15 of 7.
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