Question

(1 point) Determine the inverse modulo m of some (relatively prime) integer n. Thus, find the inverse of 16 mod 101 (i.e. an integer c such that 16c ? 1 (mod 101)). (a) Perform Euclid's algorithm on 16 and 101. 101 = 6 ( ) + = (3) + 1 (Note your answers on the second row should match the ones on the first row.) Therefore, gcd(16, 101) = 1 and 16 and 101 are relatively prime. (b) Run Euclid's algorithm backwards to write 1 = 101s + 16t for suitable integers s, t. s = t = (c) From the equation 101s + 16t ? 1 (mod 101), the multiple of 101 becomes zero (because we are considering congruence) and so we get 16t ? 1 (mod 101). Hence the multiplicative inverse of 16 mod 101 is .

          (1 point) Determine the inverse modulo m of some (relatively prime) integer n. Thus, find the inverse of 16 mod 101 (i.e. an integer c such that 16c ? 1 (mod 101)). (a) Perform Euclid's algorithm on 16 and 101. 101 = 6 ( ) + = (3) + 1 (Note your answers on the second row should match the ones on the first row.) Therefore, gcd(16, 101) = 1 and 16 and 101 are relatively prime. (b) Run Euclid's algorithm backwards to write 1 = 101s + 16t for suitable integers s, t. s = t = (c) From the equation 101s + 16t ? 1 (mod 101), the multiple of 101 becomes zero (because we are considering congruence) and so we get 16t ? 1 (mod 101). Hence the multiplicative inverse of 16 mod 101 is .

(1 point) Determine the inverse modulo m of some (relatively prime) integer n. Thus, find the inverse of 16 mod 101 (i.e. an integer c such that 16c ? 1 (mod 101)). (a) Perform Euclid's algorithm on 16 and 101. 101 = 6 ( ) + = (3) + 1 (Note your answers on the second row should match the ones on the first row.) Therefore, gcd(16, 101) = 1 and 16 and 101 are relatively prime. (b) Run Euclid's algorithm backwards to write 1 = 101s + 16t for suitable integers s, t. s = t = (c) From the equation 101s + 16t ? 1 (mod 101), the multiple of 101 becomes zero (because we are considering congruence) and so we get 16t ? 1 (mod 101). Hence the multiplicative inverse of 16 mod 101 is .

Added by Anthony H.

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