2011 AMC 8 Problems/Problem 24

Problem

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, because all odd numbers are of the form $2n+1$ where n is an integer, and 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 $2,$ so our only combination could be $2$ and $9999.$ However, $9999$ is clearly divisible by $3$ , so the number of ways $10001$ can be written as the sum of two primes is $\boxed{\textbf{(A)}\ 0}$

Solution 2 (Sort of)

One interesting way to do this is to think of $10001$ as if it's binary. Converting it to base $10$ would result in the number $17$ . Since $17$ cannot be written as the sum of two primes, the answer is $\boxed{\textbf{(A)} 0}$ .

Note: This is not a valid way to do problems like this. For example, the number $1000$ can be written as the sum of two primes in $28$ ways, but if we convert $1000$ to base ten, we would get $16$ which obviously cannot be written as the sum of two primes in $28$ ways.

Video Solution

https://youtu.be/qJuoLucUn9o by David

Video Solution 2

https://youtu.be/GqTHx0tOB4o

~savannahsolver

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

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.

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