Question

Suppose n is an integer. Select all statements below that are true:

a. n^2+n is always an even integer
b. n^2+n is always an even integer when n is even
c. n^2+n is always an even integer when n is odd
d. n^2+n is never an even integer when n is odd
e. n^2+n is is never an even integer
f. n^2+n is sometimes an even integer

Answer 1

Answers:

a. True

b. True

c. True

d. False

e. False

f. False

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Explanations:

n^2+n = n(n+1)

If n is odd, then n+1 is even, and vice versa.

Whenever you multiply an even number with an odd number, you always get an even number. This is because 2 is a factor of the overall product.

So n^2+n = n(n+1) is always even for any integer n. This makes choice A true.

Choices B and C follow immediately from this. They are more narrow examples, while choice A is more general.

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Since n^2+n = n(n+1) was shown to always be even, this means choice D is false. Choice D contradicts what choice A says. The same applies to choices E and F.