Answer the following questions completely.
1. Prove the following statement: If S = {a1, a2, ..., an} is a complete residue system modulo n, and gcd(a, n) = 1, then the set
T = {aa1, aa2, ..., aan}
is also a complete residue system modulo n.
2. Congruences respect addition, subtraction and multiplication. Let n ∈ N and let a, b, α, β ∈ Z. Suppose a ≡ α mod n and b ≡ β mod n. Then, show that
(a) a - b ≡ α - β mod n;
(b) ab ≡ αβ mod n;
3. Show that 39|53^103 + 103^53
4. Solve the following linear congruences.
(a) 25x ≡ 15 mod 29
(b) 36x ≡ 8 mod 102
(c) 34x ≡ 60 mod 98
5. Use the CRT to solve the following systems.
(a)
x ≡ 1 mod 3
x ≡ 2 mod 5
x ≡ 3 mod 7
(b)
x ≡ 5 mod 6
x ≡ 4 mod 11
x ≡ 3 mod 17
6. Solve the congruence: 17x ≡ 3 mod 210 by solving the system
17x ≡ 3 mod 2
17x ≡ 3 mod 5
17x ≡ 3 mod 7