Think of the number 7^{1999}.

Now think of it after it has been multiplied out.

What is its unit digit?

Solve problems that involve finding powers of a number

In this problem students work with powers of numbers and, as a consequence, come to understand what is happening to the numbers.

Students also see how an apparently enormous and difficult calculation can be broken down into manageable parts. The students should come to realise that there are only a limited number of unit digits obtained when 7 is raised to a power. Further, these specific digits 'cycle round' as the power of 7 increases. This cycle is 7, 9, 3, 1, 7, 9, …

The same is true of the digit in the tens place.

### The Problem

Think of the number 7^{1999}. Now think of it after it has been multiplied out. What is its unit digit?

### Teaching sequence

- Introduce the problem to the class. Check that the students understand how to raise a number to a power and how to find a power using calculator functions.
- Brainstorm ways to solve the problem.
- As the students work on the problem, either individually or in small groups, check that they are recording their solutions in ways that will enable them to look for patterns. Try to avoid
**telling**them to look for a pattern in the digits. - Share solutions.

#### Extension to the problem

How about its tens digit?

Can you find out the general pattern here. No matter what number you raise 7 to, can you tell with as little calculation as possible, what its unit digit is?

Repeat this problem with numbers other than 7.

### Solution

The answer is found when you look for patterns in the powers of 7.

7^{1 }= __7__

7^{2 }= 4__9__

7^{3 }= 34__3__

7^{4 }= 240__1__

7^{5 }= 1680__7__

7^{6 }= 11764__9__

7^{7 }= 82354__3__

7^{8 }= 576480__1__

7^{9 }= 4435360__7__

7^{10 }= 28247524__9__

The cycle for the units digit is 7, 9, 3, 1, 7, 9...

7^{1999 }Units digit = 3

#### Solution to the extension

The cycle for the tens digit is 4, 4, 0, 0, 4, 4, 0...