00:01
We are going to prove that for ab and c integer numbers, if a plus v equal c, then a times b times c is even.
00:12
So, a times b plus a times c is equal to a times b times and now we can replace c by a plus b.
00:25
So we get a times a plus b.
00:31
So the expression we want to prove is even.
00:34
Equal to this expression here.
00:37
So we have options here.
00:41
First, if a is even or b is even.
00:50
Let's see that if any of the numbers a and b is even, then this product here is even.
01:00
Okay, so if a is even.
01:06
This means that a is 2k or some k integer number.
01:21
And then replacing this expression here, we get that a times 3 times c equal a times b times a plus b.
01:32
That is 2k times b times a plus b.
01:37
And that is 2 times k times b.
01:38
And that is 2 times k times b.
01:43
Which is 2m for m equal k times b times a plus b and because here we have we have only integer numbers this product is also an integer number so a times b times c equal two times an integer number that means that a times b times c is even now the other options that was the first one the second option is b is even.
02:17
Very similar to the previous one, so b is 2k for some integer number k.
02:25
This means that a times b, which is equal to a times b times a plus b, is equal to a times 2k, replacing b by 2k times a plus b.
02:41
And now this is two times a plus b.
02:46
Sorry, i forgot the k.
02:48
Here two times a times k times a plus b and this expression here is equal to two times m for m equal a k times a plus b which is an integer number because a k and b are into the numbers so this separation here give us an integer number again and this means again that a times b times c is even.
03:24
So this is the case where a is even or b is even.
03:29
So we have the other option is neither of numbers is even.
03:34
That is both are odd.
03:37
So if a is odd and b is, it's very important to notice that this condition here is just the opposite of this one here.
03:52
It's the negation of this one...