Video: AQA GCSE Mathematics Foundation Tier Pack 1 • Paper 2 • Question 9

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.

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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.

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