Noah opened his book randomly and multiplied the numbers of the two pages he saw. If the result was 72, what were the two numbers?
We’re looking for two numbers that multiply together to make 72. We might think that any two numbers will do. One times 72 equals 72. If it didn’t matter which numbers we chose, we could just choose any pair of factors that make 72. But we need to think carefully about how Noah gets his numbers. It’s important that we understand that Noah gets his numbers from a book. Noah opens his book randomly and multiplies the numbers of the two pages that he sees.
So what do we know about these numbers? What can we say about the two factors that multiply together to make 72? They must be consecutive numbers. Consecutive numbers are numbers where one number follows on from the other. So, for example, three and four are consecutive numbers. Nine and 10 are consecutive numbers. And, of course, we’d expect the two-page numbers of two pages that are side by side to be consecutive numbers.
If we had page one and page 72 next to each other, we’d expect a whole chunk of the book to have gone missing. So which two consecutive numbers multiply together to make 72? If we don’t know the answer already, we can use our knowledge of times tables facts to try some different calculations and see where we get. Let’s start with nine and 10. We know nine multiplied by 10 is 90 which is too large. But it’s still fairly close to 72. What if we look at the next two consecutive numbers before nine and ten, eight and nine? What’s eight multiplied by nine?
We know eight multiplied by 10 or 10 times eight equals 80. So eight multiplied by nine or nine lots of eight is one lot of eight less than 80. And we know 80 take away eight equals 72. This is the number we’re looking for. If Noah opens up his book randomly and multiplies the numbers of the two pages that he sees and he gets the answer 72, then he must be looking at pages eight and nine. Eight multiplied by nine equals 72. The two numbers that Noah sees are eight and nine.