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 Write-up

Title
Units of Seven

Problem Statement

Find the units digit of 7189. Determine a general rule for finding the units digit of 7n,where n is any whole number.

Problem setup

This problem, which is very similar to other problems solved in class, deals with patterns.  For this problem, we had to figure out what digit would be in the ones place if given 7^189.

Plans to Solve/Investigate the Problem

My initial plan to solve this problem was to use an Excel spreadsheet in order to figure out the units digit of 7^189.  Unfortunately, doing so, created the following spreadsheet, which was not very helpful, seeing as though the ones digit was not listed once you got to the 189th power.  In the left hand column, I listed the exponents.  In the right hand column were the results.

 179 1.873e+151 180 1.3111e+152 181 9.1779e+152 182 6.4246e+153 183 4.4972e+154 184 3.148e+155 185 2.2036e+156 186 1.5425e+157 187 1.0798e+158 188 7.5584e+158 189 5.2909e+159

Investigation/Exploration of the Problem

Fortunately, I was able to look at the first couple of numbers to see that there was actually a pattern when looking at the ones place.  As you notice from the spreadsheet, the ones digit was 7, 9, 3, 1, (consecutively), with the pattern repeating.

 Exponent Answer 1 7 2 49 3 343 4 2401 5 16807 6 117649 7 823543 8 5764801 9 40353607 10 282475249 11 1977326743 12 13841287201 13 96889010407

After determining the pattern, I divided 189 (the exponent) by 4 (number of ones digits in the pattern), which produced an answer of 47 remainder 1.  I concluded that the remainder indicated the digit in the pattern that would fit in the ones place.  Since the remainder was 1, I concluded that this meant that the 1st number in the pattern would be the ones digit, which is 7.

Extensions of the Problem

An extension of this problem would be to experiment with other whole numbers.  For example, 5^189, or 6^189 to see if a pattern existed, and whether the units digit could be determined.

Author & Contact
Mesha Rainey
mrainey@rockdale.k12.ga.us

Link(s) to resources, references, lesson plans, and/or other materials
None

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