Math 506, Test 3 review Proofs. 1. a has a multiplicative inverse (mod m) if and only if (a, m) = 1. 2. If an ≡ 1 (mod m), then ordm (a)|n. 3. Euler’s Theorem. You may assume the permutation Lemma. 4. If (a, m) = 1 then ordm (a)|φ(m). 5. If x2 ≡ 1 (mod p) then x ≡ ±1 (mod p), where p is a prime. 6. Primality test 1: If for some integer a, with (a, m) = 1, am−1 6≡ 1 (mod m), then m is composite. 7. Primality test 2: If for some integer a with (a, m) = 1, am−1 ≡ 1 (mod m), but a(m−1)/2 6≡ ±1 (mod m) then m is composite. 8. Cancelation law. If (a, m) = 1 and ax ≡ ay (mod m), then x ≡ y (mod m). 9. If a is a nonsingular solution of f (x) ≡ 0 (mod p), then a has a unique lifting to a solution of the congruence f (x) ≡ 0 (mod p)e , for any positive e. Know the statements/definitions of the following. 1. The linear congruence ax ≡ b (mod m) is solvable if and only if d|b where d = (a, m). 2. Least complete solution set (mod m) of a congruence. Number of solutions of a congruence. 3. Complete residue system (mod m). Reduced residue system (mod m). 4. Order of a (mod m), ordm (a). 5. Eulers Theorem, Fermats Little Theorem. 6. Wilson’s Theorem. 7. RSA method for Public-Key Cryptography. (Know how it works and why it is secure.) p, q, m = pq, L = [p − 1, q − 1], de ≡ 1 (mod L). p,q,L,d are all secret. m,e are public. 8. The “Lifting Congruence” and the three possibilities. 9. Singular and nonsingular solution of a congruence (mod p). Computation 1. Solve ax ≡ b (mod m). How many solutions? 2. Find the multiplicative inverse of a (mod m). 3. Chinese Remainder Theorem. (Use the method of substitution we did in class.) 4. Decimal Expansions: Find the number of preliminary digits and the length of the repeating cycle for the decimal expansion of a/b. 5. Find the order of a (mod m). 6. Calculate ak (mod m). 7. Lift solutions of f (x) ≡ 0 (mod p) to solutions (mod p2 ) and then to (mod p3 ). 8. Solve a polynomial congruence f (x) ≡ 0 (mod 63). What steps are involved? 9. How many solutions does a polynomial congruence f (x) ≡ 0 (mod pe11 pe22 pe33 ) have given the number of solutions mod each prime power pei i .