Difference between revisions of "2002 AMC 12B Problems/Problem 11"

(Solution 2)
(Solution 2)
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If <math>A = 2</math>, then <math>A < B</math> and <math>A - B < 0</math>, which is not possible. Thus <math>B = 2</math>.
 
If <math>A = 2</math>, then <math>A < B</math> and <math>A - B < 0</math>, which is not possible. Thus <math>B = 2</math>.
  
Since <math>A</math> is prime and <math>A > A - B > 2</math>, <math>A > 3</math> and <math>A</math> can be expressed as <math>6n \pm 1</math> for some natural number <math>n</math>. Moreover for all <math>A > 5</math>, <math>A + B</math>, <math>A - B</math>
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Since <math>A</math> is prime and <math>A > A - B > 2</math>, we can infer that <math>A > 3</math> and thus <math>A</math> can be expressed as <math>6n \pm 1</math> for some natural number <math>n</math>. Moreover for all <math>A > 5</math>, <math>A + B</math>, <math>A - B</math>
  
 
== See also ==
 
== See also ==

Revision as of 14:55, 2 July 2019

The following problem is from both the 2002 AMC 12B #11 and 2002 AMC 10B #15, so both problems redirect to this page.

Problem

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

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

Solution

Solution1 1

Since $A-B$ and $A+B$ must have the same parity, and since there is only one even prime number, it follows that $A-B$ and $A+B$ are both odd. Thus one of $A, B$ is odd and the other even. Since $A+B > A > A-B > 2$, it follows that $A$ (as a prime greater than $2$) is odd. Thus $B = 2$, and $A-2, A, A+2$ are consecutive odd primes. At least one of $A-2, A, A+2$ is divisible by $3$, from which it follows that $A-2 = 3$ and $A = 5$. The sum of these numbers is thus $17$, a prime, so the answer is $\boxed{\mathrm{(E)}\ \text{prime}}$.

Solution 2

In order for both $A - B$ and $A + B$ to be prime, one of $A, B$ must be 2, or else both $A - B$, $A + B$ would be even numbers.

If $A = 2$, then $A < B$ and $A - B < 0$, which is not possible. Thus $B = 2$.

Since $A$ is prime and $A > A - B > 2$, we can infer that $A > 3$ and thus $A$ can be expressed as $6n \pm 1$ for some natural number $n$. Moreover for all $A > 5$, $A + B$, $A - B$

See also

2002 AMC 10B (ProblemsAnswer KeyResources)
Preceded by
Problem 14
Followed by
Problem 16
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
All AMC 10 Problems and Solutions
2002 AMC 12B (ProblemsAnswer KeyResources)
Preceded by
Problem 10
Followed by
Problem 12
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
All AMC 12 Problems and Solutions

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