Circle the number that is not a multiple of nine: 108, 81, 39, and 144.
There are lots of ways of approaching this problem. As the multiples of nine are the numbers in the nine times table, we could write the nine times table down. The first 12 multiples of nine are nine, 18, 27, 36, 45, 54, 63, 72, 81, 90, 99, and 108. This is because one multiplied by nine is nine, two multiplied by nine is 18, three multiplied by nine is 27, and so on.
We can see from our list that 81 and 108 are multiples of nine. Therefore, they cannot be the correct answer to this question. 39 lies between 36 and 45. As 36 and 45 are multiples of nine, 39 is clearly not a multiple of nine. Therefore, this is the correct answer.
We could continue writing the nine times table until we reach 144 to prove that this is a multiple of nine. Alternatively, we might notice that 54 plus 90 is equal to 144. Six multiplied by nine is equal to 54, and 10 multiplied by nine is equal to 90. Adding 10 and six gives us 16. Therefore, 144 four is equal to 16 multiplied by nine and is therefore a multiple of nine.
An alternate way of proving that 144 is a multiple of nine is by dividing it by nine using the bus stop method. One cannot be divided by nine. Therefore, we need to carry the one and divide 14 by nine. 14 divided by nine is equal to one remainder five. So we need to carry the five. Finally, 54 divided by nine is equal to six. Once again, we’ve proved that 144 divided by nine is equal to 16 or 16 multiplied by nine equals 144.
Whichever one of these methods we use, we can see that the number that is not a multiple of nine is 39. If we were dealing with larger numbers, we also have a rule that we could use. This states that a number is a multiple of nine if the sum of all its digits is also a multiple of nine.
For example, 108 has digits one, zero, and eight. One plus zero plus eight is equal to nine. As this is a multiple of nine, 108 must be a multiple of nine. 81 comprises of the digits eight and one. Adding these gives us a total of nine. Therefore, 81 will also be a multiple of nine. The digits in 39 are three and nine. And these add to give us 12. As 12 is not a multiple of nine, then 39 will also not be a multiple of nine. 144 has the digits one, four, and four. Once again, these three numbers sum to nine. So 144 will be a multiple of nine.
The only number out of 108, 81, 39, and 144 that is not a multiple of nine is 39.