There are 54 chips in a box. Each chip is either small or large. If the number of small chips is greater than the number of large chips by a prime number of chips, what is the greatest possible number of large chips?
I want to maximize the number of large chips, so I want to minimize the difference between the number of small and large chips. That difference has to be a prime number, so the smallest possible value is 2. That means the number of small chips is 2 more than the number of large chips, and they add up to 54. So I can write an equation: s + l = 54, where s is the number of small chips and l is the number of large chips. I can also write another equation: s - l = 2, since the difference is 2. I can solve this system of equations by adding them together, which eliminates l: 2s = 56, so s = 28. Then I can plug that into either equation to find l: l = 54 - s = 54 - 28 = 26. So the greatest possible number of large chips is 26. 