Jim has ten tiles with a different digit on each of them.

He plays around and discovers that he can make quite a lot of ten-digit numbers that are divisible by ten by using the tiles.

In fact how many can he make?

This problem investigates numbers with a given factor. Students must first be able to identify the property of a number to determine its factors. For example: A number is divisible by 10 if its last digit is 0.

Students must then be able to count in an efficient manner all the numbers with this property. Using a tree diagram with a simpler case is a suggested strategy.

This is the first of a series of three problems: Ten Tiles I, Ten Tiles II and Ten Tiles III.

Note: At The Movies, (Level 3) is a useful starting place for this problem.

Ten tiles with numbers 0 to 9 (Pieces of paper will suffice).

### Problem

Jim has ten tiles with a different digit on each of them. He plays around and discovers that he can make quite a lot of ten-digit numbers that are divisible by ten by using the tiles. In fact how many can he make?

### Teaching sequence

- Use a classroom discussion to revise the idea of factors and how you can identify numbers that have given factors.
*What are the factors of 24? 36?*

What does it mean to say that 35 is divisible by 7?

How do you know if a number is divisible by 2? 10? 5? - Pose the problem.
*Can you give me a number that is divisble by 10?*

Can you give me a ten-digit number that is divisible by 10?

How about a ten-digit number that contains each of 0, 1, 2, 3, 4, 5, 6, 7, 8, 9? - As students begin to work on the problem in their groups ask:
*How many numbers can you make using just one 0, 1 and 2, that are divisible by 10?*

How about if we use 0, 1, 2, 3 just once each? How many then?

Can you see a pattern here? - As solutions emerge, have students share and explain their approaches. Some students may have begun on the Extension problem.
- Have the students record their solutions.

#### Extension

Jessica looks at Jim’s tiles. She sees that she can work out how many two-digit numbers she can make using these tiles, that are divisible by two. How many can she make with Jim’s tiles?

### Solution

Ten-digit numbers divisible by ten: Such a number has to have a zero in the last place. Then there are 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 362880 ten-digit numbers that are divisible by ten. This is because there are 9 possible digits than can be used first. Once that is used there are 8 digits left. For any of the first 9 numbers there are 8 choices for the next number. That gives 9 x 8 = 72 possibilities so far. The pattern continues. (See also __At The Movies.__)

#### Solution to the extension:

For any number to be divisible by two the last digit has to be even. Here we have the choice of 0, 2, 4, 6 and 8. This gives us 5 choices. But we now have difficulties. Things are different if 0 is the last digit. For instance, if 5 is the last digit then we have to make sure that 0 is not the first digit. So we have to split the problem into two parts. On the other hand, we don’t have this worry if 0 is the last digit.

If 0 is the last digit, then there are 9 choices for the tens digit.

If 2, 4, 6 or 8 is the last digit then there are 8 choices for the tens digit (10 minus the 0 and minus whatever digit was used in the units position). This gives 4 x 8 = 32 numbers.

Altogether there are 41 even two-digit numbers.

(This can be done by first allowing 0 to be the first digit and then counting how many times 0 is the first digit. We can then subtract the second number – 4 - from the first - 45 – to give the answer of 41. This problem can also be solved using a tree diagram.)

(This problem can also be solved by writing out all possibilities and then counting the result. If a student does it this way that is fine. You might suggest a quicker way though.)