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Math 55, Euclidean Algorithm Worksheet Feb 12, 2013
For each pair of integers (a, b), use the Euclidean algorithm to find their gcd. Then reverse the steps of the algorithm to find integers s and t such that as + bt = gcd(a, b).
- a=254, b= 254 = 7 · 32 + 30 32 = 1 · 30 + 2 30 = 15 · 2 + 0 so gcd(254, 32) = 2. 30 = 254 − 7 · 32 2 = 32 − 1 · 30 = 32 − (254 − 7 · 32) = 8 · 32 − 254 so s = −1 and t = 8.
- a=74, b= 383 = 5 · 74 + 13 74 = 5 · 13 + 9 13 = 1 · 9 + 4 9 = 2 · 4 + 1 4 = 4 · 1 + 0
so gcd(74, 383) = 1. 13 = 383 − 5 · 74 9 = 74 − 5 · 13 = 74 − 5(383 − 5 · 74) = 26 · 74 − 5 · 383 4 = 13 − 9 = (383 − 5 · 74) − (26 · 74 − 5 · 383) = 6 · 383 − 31 · 74 1 = 9 − 2 · 4 = (26 · 74 − 5 · 383) − 2(6 · 383 − 31 · 74) = 88 · 74 − 17 · 383 so s = 88 and t = −17.
- a=7544, b= 7544 = 65 · 115 + 69 115 = 1 · 69 + 46 69 = 1 · 46 + 23 46 = 2 · 23 + 0
so gcd(7544, 115) = 23. 69 = 7544 − 65 · 115 46 = 115 − 69 = 115 − (7544 − 65 · 115) = 66 · 115 − 7544 23 = 69 − 46 = (7544 − 65 · 115) − (66 · 115 − 7544) = 2 · 7544 − 131 · 115 so s = 2 and t = −131.
- a=687, b= 687 = 28 · 24 + 15 24 = 1 · 15 + 9 15 = 1 · 9 + 6 9 = 1 · 6 + 3 6 = 2 · 3 + 0 so gcd(687, 24) = 3. 15 = 687 − 28 · 24 9 = 24 − 15 = 24 − (687 − 28 · 24) = 29 · 24 − 687 6 = 15 − 9 = (687 − 28 · 24) − (29 · 24 − 687) = 2 · 687 − 57 · 24 3 = 9 − 6 = (29 · 24 − 687) − (2 · 687 − 57 · 24) = 86 · 24 − 3 · 687
so s = −3 and t = 86.
What is the inverse of 74 mod 383?
We computed above that 1 = 88 · 74 − 17 · 383
so 1 ≡ 88 · 74 mod 383.
So by definition of inverse, 88 is the inverse of 74 mod 383.
