Let x be a positive integer. If the expression x^2 + 3x is divisible by 6, which of the fo… — GRE General Test

Let $x$ be a positive integer. If the expression $x^2 + 3x$ is divisible by $6$ , which of the following statements MUST be true?

$x$ is divisible by $3$ Answer
B
$x$ is divisible by $6$
C
$x$ is an even integer
D
$x$ is an odd integer
E
$x + 1$ is a prime number

Answer

x

is divisible by

3

Factoring

x^2 + 3x

yields

x(x+3)

. For any integer

x

, one of

x

x+3

is even and the other is odd, so their product

x(x+3)

is always divisible by

2

. For the product to also be divisible by

3

, either

x

x+3

must be divisible by

3

. Because

x

and

x+3

differ by

3

x+3

is a multiple of

3

if and only if

x

is a multiple of

3

. Thus,

x

must be divisible by

3

Step-by-Step Solution

Factor the given algebraic expression

x^2 + 3x = x(x + 3)

Factoring allows us to analyze the parity and prime factors of each component separately.

Analyze parity (divisibility by 2)

x

is even,

x+3

is odd

\Rightarrow

product is even. If

x

is odd,

x+3

is even

\Rightarrow

product is even.

The expression

x(x+3)

is guaranteed to be divisible by

2

for all integers

x

Analyze divisibility by 3

For

x(x+3)

to be divisible by

3

, at least one factor must be divisible by

3

. Since

x+3 \equiv x \pmod 3

x+3

is divisible by

3

if and only if

x

is divisible by

3

Since divisibility by

2

is always satisfied, divisibility by

6

requires

x(x+3)

to be divisible by

3

, which requires

x

to be a multiple of

3

Key Concept

Divisibility properties of factored integer expressions and modular arithmetic

Estimated Time:1m 15s

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