Noah is investigating multiples of nine: nine, 18, 27, 36, 45. When he looked at the first five numbers in the pattern, he noticed something about the sum of the digits in each number. What is it? The sum increases by two each time. The sum decreases by two each time. The sum is always nine. The sum increases by one each time. Or the sum decreases by one each time. Continue writing numbers in the pattern. What is the first multiple of nine you get to whose sum does not fit this pattern? What is special about the sum of the digits of any multiple of nine?
This is a really useful problem because it’s getting us to look at the patterns that we can see in multiples, in this case multiples of nine. And if we can see patterns in multiples, it can help us remember times tables facts. And it also helps us to identify multiples.
So we’re told then that Noah is investigating these multiples of nine. And we can see that he’s listed five of them. Nine, 18, 27, 36, and also 45. We’re then told that when Noah looked at the first five numbers in the pattern, he noticed something about the sum of the digits in each number. And we know that the word “sum” is related to addition. The sum of two numbers, or in this case two digits, is what we get when we add them together.
So in other words, Noah added the digits in each number together. And he noticed something. And we’re given five possible answers to choose from. Does the sum increase by two each time? Does it decrease by two? Is it always equal to nine? Does it increase by one each time? Or does it decrease by one each time? Let’s do exactly the same thing as what Noah has done and find the answer.
The first multiple of nine is nine. We’ve only got one digit there. So the total of this digit is just nine. 18 is made up of the digit one and the digit eight. And if we find the sum of these two digits, we’d add them together. One plus eight equals nine as well. The sum of our digits hasn’t changed. Looks like the correct answer must be that it’s always nine. Let’s keep going.
The digits in 27 are a two and a seven. And we know that two plus seven also equals nine, as does three plus six and also four plus five. Out of our five possible answers to describe the pattern, we can say that the sum is always nine. This pattern definitely works for the first five multiples of nine. But as we read the question, it gets us to think about whether it works for all of the multiples of nine.
The second part of the problem asks us to continue writing numbers in the pattern. And we’re asked, what is the first multiple of nine we get to whose sum does not fit this pattern? So clearly, this pattern doesn’t work every single time. So we’ve got up to five nines are 45.
Let’s continue counting in multiples of nine. Six nines are 54. And five plus four does equal nine. Seven nines are 63. The sum of six and three also matches the pattern. Eight nines are 72. And seven and two make nine. Nine nines are 81, still fits the pattern. Even 10 nines fits the pattern. But we can see that 11 nines, which is 99, is not going to equal nine. Nine plus nine equals 18. And so we can say that the first multiple of nine we get to whose sum does not fit the pattern is 99. So Noah needs to change the rule for his pattern slightly. And this is what the last part of our question asks.
What is special about the sum of the digits of any multiple of nine? The sum is always nine. The sum is a multiple of nine. The sum is always 18. Or the sum is a multiple of 18. Well, we can’t say that the sum is always nine because we’ve already found that when we get to 99, the sum is 18. And in the same way, we can’t say the sum is always 18 because most of the multiples of nine up to 99 added up to nine.
In fact, the only one of these answers that makes sense is that the sum is a multiple of nine. And this rule still works for larger numbers. For example, 999 is a multiple of nine. And if we add our three nines together, we don’t get nine. We don’t get 18. We actually get 27. But this still would fit the rule. The sum is a multiple of nine.
To know the interesting thing is that because it makes a multiple of nine, if we add the digits in that number together, we then get nine. So although the digits in 99 make 18, we could add one and eight. And they then make nine, and so on.
So when Noah looked at the first five numbers in the pattern, he noticed something about the sum of the digits. And the sum is always nine. We carried on writing numbers in the pattern until we got to the first multiple of nine whose sum did not fit Noah’s pattern. And that number was 99. We then altered Noah’s rule so that it made sense and was still true. The sum of the digits of any multiple of nine is that the sum is a multiple of nine.