(* You should be able to copy and paste this text file directly into Mathematica.  These notes will be interpreted as commented code. *)


(* ********* Here are the strings and numbers from the problems ******** *)

problem4text  := BRKNARJWQDBTRNBJANCQNENAHKNBCMXPB

problem5N     := 1359692821
problem5am    := 117285016
problem5bc    := 823845737

problem7cN    := 22592848554175369912787642584639796662881202104422604037797641281978785567961
problem7cphi  := 22592848554175369912787642584639796662567375207606558304355492609701088213924




(* ********* Here are some useful commands for powers mod m ******** *)

Mod[100,6]

(* This will compute 100 modulo 6, yielding 2 *)


PrimeQ[201]

(* This will test whether 201 is prime, which it is not *)


FactorInteger[143]

(* This will factor the integer 143, which is 11 times 13 *)
(* Mathematica's factorization algorithm knows how to do trial division, Pollard p-1, Pollard rho, quadratic sieving, and elliptic curve factorization, and is typically fairly fast on numbers with less than 50 digits *)


PowerMod[2, 10, 99]

(* This will compute 2^10 modulo 99.  PowerMod is efficient and will use successive squaring *)
(* Mod will often also recognize if you give it a power and internally call PowerMod, but it is better to use PowerMod directly *)



PowerMod[2, -1, 99]

(* This will compute 2^(-1) modulo 99, which is to say, the multiplicative inverse of 2 modulo 99. *)



EulerPhi[40]

(* This will compute phi(40). *)



ChineseRemainder[{1,2,3},{4,5,9}]

(* This will compute the smallest positive integer solution to x = 1 mod 4, x = 2 mod 5, x = 3 mod 9 *)





(* ********* Here are some useful other commands ******** *)

GCD[180, 250]

(* This computes the GCD of 180 and 250 *)



ExtendedGCD[180, 250]

(* This performs the extended Euclidean algorithm, and computes the GCD = 10 along with integers {x,y} such that 180x + 250y = 10 *)



PrimitiveRoot[25]

(* This calculates the smallest primitive root modulo 25 *)



MultiplicativeOrder[7, 25]

(* This calculates the order of 7 modulo 25 *)



Solve[x^3 == 1, x, Modulus -> 49]

(* This calculates all of the numbers x satisfying the equation x^3 = 1 modulo 49 *)


Reduce[x^3 == 1, Modulus -> 49]

(* This simplifies the equation x^3 = 1 modulo 49, which here will give a list of all the solutions *)