Prove that the set of positive rational numbers is countable by showing that the function K is a one-toone correspondence between the set of positive rational numbers and the set of positive integers if
K(m/n) = p 2a1 1 p 2a2 2 p 2as s q 2b11 1 q 2b21 2 q

Prove that the set of positive rational numbers is countable by setting up a function that assigns to a
rational number p/q with gcd(p, q) = 1 the base 11 number formed by the decimal representation of p
followed by the base 11 digit A, which corresponds

Show that if a b (mod m) and c d (mod m), where a, b, c, d, and m are integers with m 2, then a c
b d (mod m).
Show that if n | m, where n and m are integers greater than 1, and if a b (mod m), where a and b are
integers, then a b (mod n).

Express in pseudocode the trial division algorithm for determining whether an integer is prime.
Prove that for every positive integer n, there are n consecutive composite integers. [Hint: Consider the n
consecutive integers starting with (n + 1)! + 2.]
Sh

Suppose that a and b are integers, a 11 (mod 19), and b 3 (mod 19). Find the integer c with 0 c 18
such that
a) c 13a (mod 19).
b) c 8b (mod 19).
c) c a b (mod 19).
d) c 7a + 3b (mod 19).
e) c 2a2 + 3b2 (mod 19).
f ) c a3 + 4b3 (mod 19).

1) Prove or disprove that if a | bc, where a, b, and c are positive integers and a = 0, then a | b or a |
c.
2) Show that if n and k are positive integers, then n/k = (n 1)/k + 1.

Show that Zm with addition modulo m, where m 2 is an integer, satisfies the closure, associative, and
commutative properties, 0 is an additive identity, and for every nonzero a Zm, m a is an inverse of a
modulo m
Show that the distributive property of mul

Show that if a and b are positive integers, then ab = gcd(a, b) lcm(a, b). [Hint: Use the prime
factorizations of a and b and the formulae for gcd(a, b) and lcm(a, b) in terms of these factorizations.]
How many divisions are required to find gcd(21, 34) u

. Adapt the proof in the text that there are infinitely many primes to prove that there are infinitely many
primes of the form 3k + 2, where k is a nonnegative integer. [Hint: Suppose that there are only finitely
many such primes q1, q2,.,qn, and consider