Powers of Seven
Solve problems that involve finding powers of a number
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 students 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 students 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.
The Problem
Think of the number 71999. 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 them 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?
Comment: This problem can be repeated from time to time with numbers other than 7.
Solution
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
Solution to the extension
The cycle for the tens digit is 4, 4, 0, 0, 4, 4, 0...
| Attachment | Size |
|---|---|
| Powers.pdf | 34.17 KB |
| PowersMaori.pdf | 45.94 KB |
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