### Instruction: What is the greatest whole number that MUST be a factor of the sum of any four consecutive positive odd numbers?### Response: I need to find a common factor of the sum of any four consecutive positive odd numbers, regardless of which ones they are.
I could try some examples and look for a pattern, but I also want to have a more general argument.
I notice that any odd number can be written as 2k+1, where k is a whole number.
So, if I have four consecutive odd numbers, they can be written as 2k+1, 2k+3, 2k+5, and 2k+7.
The sum of these numbers is 8k+16.
Now, I want to find the greatest possible factor of 8k+16 that does not depend on k.
I see that 8 is a factor of 8k+16, regardless of the value of k.
Is 8 the greatest possible factor?
I can check by dividing 8k+16 by 8 and seeing if the quotient is always a whole number.
The quotient is k+2, which is a whole number for any value of k.
Therefore, 8 is the greatest whole number that MUST be a factor of the sum of any four consecutive positive odd numbers.