cs2022a16quiz06.pdf - Ma2201/CS2022 Quiz 0110 Discrete...

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Ma2201/CS2022 Discrete Mathematics A Term, 2016 Quiz 0110 Print Name: Sign: 1. ( 3 pts ) Find the multiplicative inverse of 23 in Z 101 . The obvious ones in Z 101 come from 1 102 = (2)(51) mod 101 and 1 ≡ - 100 = (2)( - 50) = (4)( - 25) = (5)( - 20). But we’re stuck with 23, so we use the Euclidean Algorithm: 101 = 4 · 23 + 9 23 = 2 · 9 + 5 9 = 1 · 5 + 4 5 = 1 · 4 + 1 We add these with coefficients to remove the inner remainders: 5 : 101 = 4 · 23 + 9 2 : 23 = 2 · 9 + 5 - 1 : 9 = 1 · 5 + 4 1 : 5 = 1 · 4 + 1 Giving (5)(101) + (22)(23) = 1 so the multiplicative inverse of 23 is 22, which checks since (22)(23) = 460 + 46 = 506 = 505 + 1. 2. ( 3 pts ) Suppose 17 m + 4 39 m - 7 mod 101. Find m . Gathering terms we get 11 22 m mod 101, and we need the multiplicative inverse of 22 modulo 101, which we have from the previous problem: (23)(11) (23)(22) m m mod 101 . So m = (11)(23) = 230 + 23 = 253 = 202 + 51 51 mod 101.
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