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Notation: mod and % denote the modulo operator for modular arithmetic
Notation: n^x means n raised to the power of x
x ≡ y reads "x is congruent to y"
Parity refers to the even/odd property of an integer; e.g., 2 has even parity
n is even if and only if n = 2k, for some k.
n is odd if and only if n = 2k + 1, for some k.
(k)(k+1) is even and k + (k+1) is odd.
(2k+1)(2k) is even and (2k+1)+(2k) is odd.
m mod n is the positive remainder of m/n.
m mod n = m - nq, for some q >=0.
If m < n, then m mod n = m; e.g., 4 mod 10 = 4.
m mod 1 = 0; e.g., 4 mod 1 = 0.
If m and n are congruent mod x then (m mod x) = (n mod x).
If m and n are congruent to 0 mod d, then m=dk and n=dj, for some k and j.
If m and n are congruent to each other mod 2, then m and n share parity.
(x + y) mod m = ((x mod m) + (y mod m)) mod m.
(xy) mod m = ((x mod m)(y mod m)) mod m.
If n is not divisable by 3 then n=3k+1 or n=3k+2, for some k.
If n is composite then n = pq, for some p and q > 1.
If n is prime then there are no factors in n other than 1 and n.
If n is rational then n can be expressed as x/y, for
y > 0 and where x and y are relatively prime.
x and y are relatively prime if they share no common factor other than 1.
If d is a factor of n, then n = dk, for some k.
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