### Video Transcript

Joanna states, “five is odd and
seven is odd, so the product of any multiple of five and any multiple of seven must
also be an odd.“ Is she correct? Give an example to support your
answer.

First, let’s think about what we
know about even and odd multiplication. An odd number times an odd number
equals an odd number. An example of this would be three
times five equals 15. Then, we have the odd times even
case, which equals an even number. We see this in three times two
equals six. The last case, even number times an
even number equals an even number, an example being four times two equals eight.

If you didn’t remember the rules,
you could consider examples of each scenario to help you remember. Now we’ve considered the even and
odd multiplication, we need to ask the question “is every multiple of five and every
multiple of seven odd?“

A multiple is what we get when we
multiply a number by an integer. Right here, we should note that
integers can be both even and odd. Let’s list some of the multiples of
five: five times one, where one is the integer we’re multiplying five by, equals
five; five times two, where two is the integer we’re multiplying five by, which
equals 10; five times three equals 15; five times four equals 20.

We’re already noticing that five
has some even-numbered multiples. The first multiple of seven, seven
times one, equals seven, seven times two equals 14, seven times three equals 21, and
seven times four equals 28. Seven also has even-numbered
multiples.

Joanna’s claim is that multiplying
any multiple of five by any multiple of seven will always be odd. Imagine we choose 10 as our
multiple of five and seven as our multiple of seven. 10 times seven equals 70. We know that 10 is a multiple of
five, it’s five times two, and that seven is a multiple of seven, seven times one,
that this two makes five times two makes 10 an even number multiplied by a multiple
of seven, seven which is an odd number and they produce an even number. Although 10 and seven are both
multiples of five, 70 is not odd.

This is a counterexample. It’s a place where Joanna’s
statement is false. And we only need one counterexample
to make the whole statement false. Joanna is not correct because
multiplying seven or five by an even number produces an even multiple, which will
yield an even final product. Remember that in this question,
it’s not enough just to answer is she correct. You also need to support your
answer with an example, which we’ve done here.