Mathematics

Problem

A two-column proof shows that the sum of two odd integers is even: 1.

x = 2k + 1

y = 2l + 1

Definition of odd 2.

x + y = 2k + 1 + 2l + 1

Addition 3.

x + y = 2(k + l + 1)

Factorisation 4.

x + y

is divisible by

2

Definition of even Which paragraph proof correctly combines all four steps in proper sequence? a) Assume first that

x + y

is divisible by

2

. Because

x + y = 2k + 1 + 2l + 1 = 2(k + l + 1)

, we can write

x = 2k + 1

and

y = 2l + 1

. Hence

x

and

y

are odd. b) Suppose

x = 2k + 1

and

y = 2l + 1

. Adding gives

x + y = 2k + 1 + 2l + 1 = 2(k + l) + 1

. Because this expression contains a factor of

2

x + y

is even. c) Let

x

and

y

be odd integers so that

x = 2k + 1

and

y = 2l + 1

for some integers

k,l

. Then

x + y = 2k + 1 + 2l + 1 = 2(k + l + 1)

, which is divisible by

2

. Therefore

x + y

is even. d) Suppose

x = 2k + 1

and

y = 2l + 1

. Their product is

(2k + 1)(2l + 1) = 2(2kl + k + l) + 1

, which shows

xy

is odd; therefore

x + y

must be even.