2011 AMC 8 Problems/Problem 24
Contents
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.
Solution 3(Simple)
First, we noticed that 10001 is equal to 5000+5001, if you subtract n to 5000 and add n to 5001, you always get an even number, even number is never a prime number except 2. We try 2 and 9999 but we can see 9999 is divisible by 3 and 9 clearly. So the answer is
Video Solution
https://youtu.be/qJuoLucUn9o by David
Video Solution 2
~savannahsolver
See Also
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 |
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