2011 AMC 8 Problems/Problem 24

Problem

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

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 where n is an integer, and all even numbers are of the form where m is an integer. and is an integer because and 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

Solution 2 (Sort of)

One interesting way to do this is to think of as if it's binary. Converting it to base would result in the number . Since cannot be written as the sum of two primes, the answer is .

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

Video Solution

https://youtu.be/qJuoLucUn9o by David

Video Solution 2

https://youtu.be/GqTHx0tOB4o

~savannahsolver

See Also

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