If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains ***.kastatic.org** and ***.kasandbox.org** are unblocked.

Main content

Current time:0:00Total duration:3:40

- We are told the product
of two even numbers is even. And that is true for any two even numbers. The product of an odd number
and an even number is even. That is also true for any
two numbers, two integers. The product of p and r is odd. And they say if p and r are integers, then, according to the statement above, which of the following
is a valid conclusion? And, you only need to pick one of these. So pause this video and
see if you can answer that. Okay, now lets do it together. I want us to quickly go
through each of these choices and see if these can
be a valid conclusion. So, the first statement
says, only p is odd. So only p is odd means
that r would be even. So, let's just use two examples, let's say that p is five. So, I'm just picking an
arbitrary odd number. And lets make r, four. So what's the product of five and four? Fives times four is going to
be equal to 20, which is even. So if only p is odd, the product is going to be an even number. But they say that the
product of p and r is odd, so, this cannot be true. Because here the product is
even, so we can rule that out. And another way to think about it, only p is odd, means that you
have a product of an even, and an odd because an r
would have to be even. Well, that's the second case over here and they tell us your
product is going to be even. We're in a situation
where the product is odd, so once again we rule that out. Choice B only r is odd, so it's the same idea, if
r is odd then p is even. But we could if we wanted to, try out values we could say
let's make r equal to five, or three, or seven, or
nine, some odd number and let's make p even which would be four and once again you get five times four is be equal to 20. The product would be even, not odd. So, this is not a valid conclusion. Once again, if you have an odd and an even that's this second scenario up here, where you know the product
will be even, not odd. Either p or r is both, but, Sorry. Either p or r is odd, but not both. Well, that's these other two cases. If p is odd but they're not both odd, then we're in choice A scenario, and if r is odd but p isn't, well we're in choice B scenario and so, I would rule that one out as well. This is going to be a even times an odd, which is the second scenario
where the product is even, they tell us that and you can try out some values to verify that. But, we want to figure out
statements about p and r, so, that we know so that we
know there product is odd. Both p and r are odd. So, you could try out
some values if you like, to give you confidence here. So, lets say three times five,
that's 15, that checks out. So, this is odd, that is
odd, the product's odd, you do seven times nine is
equal to 63, that checks out. You could keep trying this out but this ones holding up pretty well. In other videos we could try and prove it, but this one's looking pretty good, I'll put a smiley face there. Both p and r are even. Well, they tell us that if the product of
two even numbers is even. So, this is going to be an even, this is going to be an even
product right over here. They told us that the
product of p and r is odd, so we can rule that out. You could also try some examples here, you could say two times four,
well two times four is eight. Even times even is an even. But p times r has to be odd, so we can rule that one out as well. So, we can feel very good about choice D.