Unformatted text preview: A summary of the maths you’ve learnt from Week 1 to Week 4 Written by: Vicky Mak (Burwood) [email protected] August, 2008 1 The maths we’ve learnt so far 1. Do you know how to solve ax + by = gcd(a, b)? The Extended Euclidean Algorithm 2. What can say about a and b if you can ﬁnd x, y integers such that ax + by = 1? This means that gcd(a, b) = 1, that is, a and b are coprime. 2 The maths we’ve learnt so far 3. Do you know how to ﬁnd the inverse of a( mod n) for a, n coprime? Easy, solve as + nt = 1, then a−1 ≡ s( mod n). 4. How about solving ax ≡ c ( mod n)? Read Modular Not hard either, but there are 3 cases. Airthmetic slides I put on DSO. 3 The maths we’ve learnt so far 5. You need to know the Chinese Remainder Theorem. See Week 3 Lecture Slides and the additional examples inserted within the powerpoint slides. You need to know how to state the general case and of course how to use it for solving 2 simultaneous congruences. 6. The Fermat’s Little Theorem A must know! If p is prime and that p does not divide a, then ap−1 ≡ 1( mod p). 4 The maths we’ve learnt so far 7. Euler’s Theorem. Also a must know. If gcd(a, n) = 1, then aφ(n) ≡ 1( mod n), for φ(n) the number of numbers from 1 to n − 1 that are coprime to n. 6. Primitive Roots and Square Root Modulo n What are these? See Lecture slides from Week 4. 5 ...
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 Spring '10
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 Number Theory, Prime number, Euclidean algorithm, [email protected], Vicky Mak

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