week 2: number theory practice questions pending

remainders/residues/residue classes

Q1. If a = b mod m and c = d mod m, then:
Prove that:
(1) a +- c = b +- d mod m
(2) ac = bd mod m
(3) ax + cy = bx + dy mod m

Q2. if a = b mod m, pr. th. a^n = b^n mod m.

Q3. If a = b mod m then a = b mod d if d | m.

Q4. 
Remainder of 13^73 + 14^3 mod 11.

Q5.
Pr. th.
ax = ay mod m iff x = y mod (m/gcd(a,m))
S5.
m = k.q1, a = k.q2 where k = gcd(a,m)
m/k = q1
x = y mod q1 => ax = ay mod q1 => ax = ay mod m since q1| m.
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