2011 AMC 8 Problems/Problem 24

In how many ways can $10001$ be written as the sum of two primes?

$\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }3\qquad\textbf{(E) }4$

Solution

For the sum of two numbers to be odd, one must be odd and the other must be even.

Proof:

All odd numbers are of the form $2n+1$ where n is an integer. All even numbers are of the form $2m$ where m is an integer. $2n + 1 + 2m = 2m + 2n + 1 = 2(m+n) + 1$ and $m+n$ is an integer because $m$ and $n$ are both integers.

The only even prime number is  so our only combination could be  and  However,  is clearly divisible by  so the number of ways  can be written as the sum of two primes is

2011 AMC 8 (Problems • Answer Key • Resources)
Preceded by Problem 23	Followed by Problem 25
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2011 AMC 8 Problems/Problem 24

Solution

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