If p, p + 2, and p + 4 are all prime numbers, what is the value of p^2 + 5? — GRE General Test

If $p$ , $p + 2$ , and $p + 4$ are all prime numbers, what is the value of $p^2 + 5$ ?

A
6
B
9
14Answer
D
19
E
30

Answer

The value of

p^2 + 5

is 14.

For any three consecutive odd integers

p

p + 2

, and

p + 4

, exactly one of them must be a multiple of 3. Because all three expressions represent prime numbers, the term divisible by 3 must be equal to 3 (since 3 is the only prime divisible by 3). Setting

p = 3

gives

p + 2 = 5

and

p + 4 = 7

, both of which are prime. Any choice of

p > 3

forces either

p + 2

p + 4

to be a multiple of 3 greater than 3, making it composite. Thus

p = 3

is uniquely determined, and evaluating

p^2 + 5

gives

3^2 + 5 = 14

Step-by-Step Solution

Analyze the possible remainders when the prime

p

is divided by 3.

Any positive integer

p

can be written in one of three forms:

3k

3k + 1

, or

3k + 2

for some integer

k

Dividing any integer by 3 leaves a remainder of 0, 1, or 2.

Test each remainder case for the expressions

p

p + 2

, and

p + 4

p = 3k + 1

, then

p + 2 = 3k + 3 = 3(k + 1)

, which is divisible by 3. If

p = 3k + 2

, then

p + 4 = 3k + 6 = 3(k + 2)

, which is divisible by 3.

Among any three consecutive odd numbers of the form

p, p+2, p+4

, exactly one of them must be a multiple of 3.

Deduce the unique value of

p

The only way all three numbers

p

p + 2

, and

p + 4

can be prime is if the one divisible by 3 is equal to 3 itself, which forces

p = 3

The only prime number divisible by 3 is 3 itself; any larger multiple of 3 is composite.

Evaluate the target expression

p^2 + 5

using

p = 3

3^2 + 5 = 9 + 5 = 14

Substitute the uniquely determined value

p = 3

into the given algebraic expression.

Key Concept

Divisibility properties of consecutive odd integers and prime number definitions

Estimated Time:1m 30s

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