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 .