Mathematics

Problem

In a structured proof that the product of two odd integers

x

and

y

is odd, the following steps are recorded: (i)

x = 2k + 1

y = 2l + 1

for integers

k, l

. (ii)

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

. (iii) _____ (iv) Therefore

xy

is of the form

2m + 1

, so

xy

is odd. Which statement best completes Step (iii)? a)

xy = 4kl + 2k + 2l

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

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

xy

is divisible by

2

Intuitive explanation

The key insight is recognizing that you need to expand the product and then factor out a 2 to reveal the form $2m + 1$ . Step (iii) should show the algebraic manipulation that transforms $(2k + 1)(2l + 1)$ into an expression where you can clearly identify the even part and the remainder of 1—this is what proves the result is odd. Think of it as "making the structure visible" so step (iv) can conclude.

Intuitive explanation

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