2002 AMC 10B Problems/Problem 15

Problem

The positive integers $A$ , $B$ , $A-B$ , and $A+B$ are all prime numbers. The sum of these four primes is

$\mathrm{(A) \ } \text{even}\qquad \mathrm{(B) \ } \text{divisible by }3\qquad \mathrm{(C) \ } \text{divisible by }5\qquad \mathrm{(D) \ } \text{divisible by }7\qquad \mathrm{(E) \ } \text{prime}$

Solution

The sum is $A+B+A-B+A+B=3A+B$ . Since $A$ , $A-B$ , and $A+B$ are all prime, they must all be odd, so $B=2$ . A quick check gives $A=5$ . Hence, the sum is $17$ , which is prime. $\mathrm{ (E) \ }$

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2002 AMC 10B Problems/Problem 15

Problem

Solution