Integers and Modular Arithmetic

Pay close attention to the definition of t|s

Determine gcd(24* 32*5*72,2*33*7*11) and lcm(23*32*5, 2*33*7*11).

Show that if a and b are positive integers, then ab=lcm(a,b)*gcd(a,b).

Determine 71000 mod 6 and 61001 mod 7.

If a and b are integers and n is a positive integer, prove that a mod n = b mod n if and only if n divides

a-b.

Let n and a be positive integers and let d= gcd(a,n). Show that the equation ax mod n=1 has a solution

if and only if d=1. (This exercise is referred to in Chapter 2.)

Prove: If a, b, c Z, the set of integers, and a|b and b|c, then a|c.

That is, prove that divisibility is transitive.

Let b, b, a Z, and let a 0.

Let b = aq + r,

b = aq + r where 0

Since 10 = 3 mod 7, 100 = 2 mod 7, and 1000 = 6 mod 7, prove that 7|(1000a + 100b + 10c + d) if

and only if 7|(6a + 2b + 3c + d). Using this theorem, show that 7|8638.

Give an example where ac = bc mod m, c 0 mod m, and a b mod m.

Induction and Equivalence Relations

Prove that there are infinitely many primes. (Hint: Use Exercise 20.)

For every positive integer n, prove that 1+2++n=n(n+1)/2.

Use the Generalized Euclids Lemma (see Exercise 26) to establish the uniqueness portion of the

Fundamental Theorem of Arithmetic.

What is the largest bet that cannot be made with chips worth $7.00 and $9.00? Verify that your

answer is correct with both forms of induction.

48.

49.

Explain the difference between the First Principle of Mathematical Induction and the Second Principle

of Mathematical Induction, using problems 22 and 28 as examples. Explain why the Second Principle

is sometimes called the Strong From of Mathematical Induction.

Prove: 1 + 3 + 5 + ... + (2n + 1) = (n+1)2 for all integers n >0.

What is wrong with the following proof?

Theorem: All horses have the same color.

Proof: Let P(n) be the proposition that all horses in a set of n horses are the same color.

For n=1, P(n) is clearly true. Let n be an integer for which P(n) is true. We want to prove that P(n+1) is

true.

Let H1, ..., Hn+1 be n+ 1 horses in a set of n+1 horses. Consider S = { H1, ..., Hn}. By the inductive

assumption, all the horses in S are the same color. Now, replace Hn in S, with Hn+1 to form the set S,

which also is a set of n horses, and hence all the same color. Since Hn is the same color as H1, and Hn+1

and H1 are the same color, all n+1 horses are the same color. This concludes the proof.

a.) Define a relation R for points on the surface of the Earth as follows:

p~q if p and q have the same latitude. Is ~ an equivalence relation? If so, prove it and describe the

equivalence classes. If not, give a counterexample.

b.) Same as part a, but with longitude instead of latitude.

Define a relation ~ on the integers as follows: a~b if 5|(a-b). Prove this is an equivalence relation.

Describe the equivalence classes in the partition. Can you phrase this in terms of modular arithmetic?

Define a relation on the complex numbers as: a + bi ~ c + di if and only if a2 + b2 = c2 + d2.

Prove this is an equivalence relation and describe the equivalence classes in the partition.