14 Chapter 2

Try proof by cases. After proving one special case, look for the remaining cases.

Sometimes one can prove a statement by totally separate arguments in the various

possible cases.

Exercises

2.1 Prove that the sum, difference, and product of two even numbers is even. Similarly, prove that

the sum and difference of two odd numbers is even, but the product of odd numbers is odd.

2.2 True or false: if the sum of one pair of positive integers is larger than the sum of another pair,

then the product also is larger.

2.3 True or false: if the product of one pair of positive integers is larger than the product of another

pair, then the sum also is larger.

2.4 Prove that the number of ways to choose two things from a set of n things is n(n − 1)/2. In

other words,



n

2



= n(n −1)/2.

2.5 Prove that the product of k consecutive integers is always a multiple of k.

2.6 Prove that the product of any three consecutive integers is a multiple of 6. Conclude that n

3

−n

is a multiple of 6 for every integer n.

2.7 Prove that the product of any four consecutive positive integers is a multiple of 24.

2.8 Generalize exercise 2.7 to the product of five consecutive numbers, and to six, and so on.

Formulate a conjecture concerning the product of any n consecutive positive integers being a

multiple of some number depending on n. Can you prove it?

2.9 State and prove a theorem concerning positive integers k for which the product of any k − 1

consecutive positive integers is a multiple of k.

2.10 Can you criticize the following argument? Claim. n

3

−n is always even for any natural number

n. Proof. Consider an n ×n × n cube, consisting of n

3

many small cubes. Remove the n small

cubes on the long diagonal through the cube, so there are n

3

−n many cubes remaining. Since

the cube is separated into two identical and symmetric pieces by the main diagonal, one on

each side, it must be that n

3

− n is even.

Credits

The essential idea of this chapter was adapted from a treatment of Benjamin Dickman, who

first proposed the n

2

− n example on MathOverflow, Dickman (2013), and later developed

it more fully in his article, Dickman (2017), written for mathematics educators.