If x is any integer, which of the following expressions must be divisible by 2? — GRE General Test

If $x$ is any integer, which of the following expressions must be divisible by $2$ ?

The expression

x^2 + x + 4

must be divisible by

2

for all integer values of

x

The expression

x^2 + x + 4

can be rewritten as

x(x + 1) + 4

. For any integer

x

, either

x

x + 1

is even, so their product

x(x + 1)

is always even (a multiple of

2

). Adding

4

(which is also even) to an even number always yields an even number. Therefore,

x^2 + x + 4

is divisible by

2

for all integer values of

x

Rewrite the expression by factoring out

x

from the first two terms.

x^2 + x + 4 = x(x + 1) + 4

Grouping terms isolates the product of two consecutive integers.

Analyze the parity of the product

x(x + 1)

Because

x

and

x + 1

are consecutive integers, one of them must be even, making the product

x(x + 1)

an even integer.

The product of any integer and its immediate successor is always even.

Add

4

to the even product

x(x + 1)

The sum of an even integer and an even integer (

4

) is always an even integer.

Adding an even integer to an even integer preserves divisibility by

2

Parity of consecutive integer products and integer divisibility rules

Question