2002 AMC 10B Problem 15

Question

The positive integers

A,

B,

A - B,

and

A + B

are all prime numbers. The sum of these four primes is

even

divisible by

3

divisible by

5

divisible by

7

prime

Answer: E

Concepts:prime parity

Difficulty rating: 1480

Small hint:

A - B

and

A + B

have the same parity, and both are prime

Big hint:

This forces

B = 2,

so

A - 2,

A,

and

A + 2

are three primes in a row

Solution:

The numbers

A - B

and

A + B

differ by

2B,

so they have the same parity. Being prime, they must both be odd, which forces

A

and

B

to have opposite parity.

Since

2

is the only even prime, either

A=2

or

B=2.

The first case is impossible because the positive prime

B

would make

A-B<0.

Hence

B=2.

Now

A - 2,

A,

and

A + 2

are three primes. One of any three integers spaced

2

apart is divisible by

3,

so that member must itself be

3.

The only positive possibility is

3,

5,

7.

The four primes are

2,

3,

5,

7,

and their sum is

17,

which is prime.

Thus, the correct answer is E.

2002 AMC 10B Problem 15

Answer: E

Difficulty rating: 1480

Small hint:

Big hint:

Solution:

Problem 15 in other years