Before someone comes in and says 0 is neither even nor odd and tries to make an explanation from that I will let you guys know that it is a fact that 0 is even.
I don't know what argument they're trying to make. The question as you stated it asks if there exists a combination of 4 integers(one of them being 0) where you could make the sum and product an even number. The way you phrase it asks if there is a possible combination where the condition is satisfied--not if all possible combinations of integers satisfy the condition. I think that's the mistake the author is making. That and the poor guy thinks 0 is odd. What he is saying is not even true(pun intended)!
To provide an example: 0,1,2,3 sums to an even number and the product is even. There is your example right there.
Let's see if we can do this rigorously to map out all the possible conditions. In the system I'm creating any even number counts for 0 and any odd number counts as 1. You can check this system is correct by checking all the rules of summing even and odd numbers on the system and see if its the same for the set of real numbers(the rules being even+odd=odd, even+even=even, odd+odd=even). If the sum of my 0-1 system is even then the sum of the numbers is even. If the sum of the 0s and 1s are odd then the sum is odd.
1 1 1 1
1 1 1 0
1 1 0 0
1 0 0 0
0 0 0 0
Those are our possible representations of even and odd integers. So off the bat there are no combinations where there are 1 or 3 odd numbers in the set because they won't satisfy the 'sum to an even number' condition. Of the remaining we have 4,2, or 0 odd numbers. Here are the rules for products of multiplication:
odd*even=odd
even*even=even
odd*odd=even
If we have 4 odd numbers then we get an even number when we multiply them because (odd*odd)*(odd*odd)=even*even=even.
If we have 2 odd and 2 even we get an even number when we multiply then because odd*odd*even*even = even^3 = even.
If we have 0 odd numbers then the sum is even and the product is even because even*even*even*even=even.
So any situation where you have either 4, 2, or 0 odd numbers both conditions are satisfied.
Ah ****, I'm late to pickup my new bike:
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