The product of the positive difference of eight integers whose sum is 28 will always be divisible by what power of 6?
A number is divisible by 6 if 2 and 3 appear as prime factors in it. In the difference of two numbers, 2 can only appear as a prime factor if both numbers are even or both are odd. The worst-case scenario is when 4 numbers are even and 4 are odd. In this case, only the difference of numbers of the same parity will yield an even number, meaning it will be divisible by the twentieth power of 2. From the perspective of divisibility by 3, the numbers can be classified into types $3x, 3x-1, 3x-2$. These cover all possible integers. The difference of two numbers is divisible by 3 if the two numbers belong to the same type. The worst-case scenario is when 3-3-2 numbers belong to the same group. In this case, 3 appears only twice as a prime factor. Therefore, the product of the differences is at least divisible by the seventh power of 6. (János Korda)