# Chapter 1: Introduction to Software
# 
# 
# The main purpose of this section is to give you a chance to practice
# using Maple.
# It has several functions which are relevant to this section. For
# making computations
# in the integers modulo n there is a built-in function modp. Thus
# 
> modp(25+87,13);

                                  8

# 
# and
# 
> modp(2^12,7);

                                  1

# 
# For any real number a the function evalf(a,m) will compute the first m
# digits of a.
# For example,
# 
> evalf(1/7,20);

                        .14285714285714285714

# 
# So we see that the period of 1/7 is 1,4,2,8,5,7  as discussed above.
# Similarly
# 
> evalf(1/19,40);

              .05263157894736842105263157894736842105263

# 
# Thus the period of 1/19 is  0,5,2,6,3,1,5,7,8,9,4,7,3,6,8,4,2,1  which
# has length 18. And
# 
> modp(10^18 - 1, 19);

                                  0

# 
# but for smaller k , 
> 10^k-1;#   is not congruent to 0 mod 19. We can do these calculations
# for all the primes less than 30 say, and tabulate the results. They
# confirm theorem 1.3.
# Interestingly
# 
# 1001 = 
> 10^3;#  + 1 = 7*11*13  ,
# 
# so that
# 

> 10^3;# + 1  =   0  mod 7 .
# 
# Similarly,
# 
> modp(10^9 + 1, 19);

                                  0

# 
# What do you think is going on? Experiment a bit and try to make a
# conjecture.
# 
# Given a pair of integers a and b, the function igcdex will compute (a,
# b) and
# integers s and t such that (a, b) = as + bt . For example,
# 
> igcdex(57,21,'s','t');

                                  3

> s;t;

                                  3


                                  -8

# 
# This means that  3 = 3*57 - 8*21 . If we want to find the
# multiplicative
# inverse of 105  in 
> F[197];# , we can enter
# 
> igcdex(105,197,'s','t');

                                  1

> s;t;

                                 -15


                                  8

# 
# So 1 = -15*105 + 8*197  and  -15 = 182 is the multiplicative inverse
# of 105 .
# 
# In the package NumberTheory`NumberTheoryFunctions` there is a function
# called
# chrem which will solve two such congruences simultaneously. In fact it
# will solve a
# system: you simply enter
# 
> chrem([b1, ..., br], [n1, ..., nr]);
# 
# So in our example:
# 
> chrem([14,6],[24,31]);

                                 254