Armstrong Numbers

Grade Level: 
High School
Subject: 
STEM

Overview
An Armstrong number is an n-digit number that is equal to the sum of the nth powers of its digits. In this lesson, students will explore Armstrong numbers, identify all Armstrong numbers less than 1000, and investigate a recursive sequence that uses a similar process. Throughout the lesson, students will use spreadsheets or other technology. 

Goals/Objectives
Students will:

* Formulate a definition of Armstrong numbers based on several examples.

* Identify all Armstrong numbers less than 1000 using appropriate technology.

* Reasonably explain why the sequence of Armstrong numbers is finite. 

Time Required: 2 periods 

Materials
Computer and Internet connection

Armstrong Numbers Spreadsheet <https://www.istemnetwork.org/resource/educational/lessondetail.cfm?lessonid=149>

Armstrong Iteration Spreadsheet Strong Arm Iteration Activity Sheet <http://illuminations.nctm.org/Lessons/Armstrong/Armstrong-AS-StrongArm.pdf>


Procedure
Explain to students that they will be implementing the following process for several numbers:

1. Raise each digit to a power equal to the number of digits in the number. For instance, each digit of a four-digit number would be raised to the fourth power; each digit of a five-digit number would be raised to the fifth power; and so on.

 

2. Add the results. Demonstrate this process for 123—because 123 is a three-digit number, raise each digit to the third power, and add: 13 + 23 + 33 = 36.

 

If necessary, show other examples. For the complete procedure please follow the lesson plan link <http://illuminations.nctm.org/LessonDetail.aspx?id=L648>

Closure
Have students consider the following:

1. If you begin with 123, the sequence reaches 153, and then it begins to repeat. That is, 13 + 53 + 33 = 153. What other three-digit numbers will eventually reach 153 and begin to repeat? Is there a pattern to the numbers that reach 153? [There are many numbers that reach 153 and then repeat; some of the numbers are 135, 213, 369, 423, 546, 678, 775, 819, and 972. All multiples of 3 eventually reach 153.]

 

2. Other than 153, what other numbers are reached when this process is applied? [The other three-digit Armstrong numbers are occasionally reached. For instance, 124 eventually leads to 370; 551 leads to 371; and 740 leads to 407.]

 

3. What other interesting things did you notice during this investigation? If possible, explain why these interesting things happened. [Several numbers lead to a cycle of repetition rather than to an Armstrong number. 136 leads to 244 which leads back to 136. Cycles of three numbers also occur.] 

Assessment

1. Require students to write a journal entry that:

a. identifies all of the Armstrong numbers less than 1000, and

b. explains how they know that they’ve found all of them.

 

2. Review students’ spreadsheets or computer programs to determine their method for finding Armstrong numbers. Even if students were not able to find all Armstrong numbers less than 1000, rate the work completed.