Problem
Think of the number 7¹⁹⁹⁹. Now think of it after it has been multiplied out. What is its unit digit?

What is this problem about?
This problem has two purposes. First it is an opportunity to practice taking powers of numbers and to understand what that process is all about. Secondly it gives children the chance to see how a seemingly enormous and apparently difficult calculation can be broken down to something that is within everyone’s reach. The children should come to realize that there are only a limited number of unit digits obtained when 7 is raised to a power. Further that these specific digits cycle round as the power of 7 increases. This cycle is 7, 9, 3, 1, 7, 9, …

The same is true for the tens digit. There is a cycling round of numbers here too.

Achievement Objectives
Number (Level 4)
- explain the meaning and evaluate powers of whole numbers.

Mathematical Processes
- devise and use problem solving strategies (guess, be systematic, look for a pattern)

Resources
Calculators
Blackline master of the problem (English)
Blackline master of the problem (Maaori)

Specific learning outcomes
The children will be able to:
- solve problems that involve finding powers of a number

1. Introduce the problem to the class. Check that the children understand how to raise a number to a power and how to find them using calculator functions.
2. Brainstorm ways to solve the problem.
3. As the children 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.
4. 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?
Comment: This problem can be repeated from time to time with numbers other than 7.

Solution to the problem
The answer is found when you look for patterns in the powers of 7.
7¹ = 7
7² = 49
7³ = 343
7⁴ = 2401
7⁵ = 16807
7⁶ = 117649
7⁷ = 823543
7⁸ = 5764801
7⁹ = 44353607
7¹⁰ = 282475249

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