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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

Given any whole number power of 7, is there a rule for determining the units or ones digit?

Plans to Solve/Investigate the Problem

I will begin with smaller powers of 7 and look for a pattern. I will use the on-computer calculator for as a computational tool.

Investigation/Exploration of the Problem

70 = 1

71 = 7

72 = 49

73 = 343

74 = 2401

75 = 16807

76 = 117649

77 = 823543

78 = 5764801

79 = 40353607

710 = 282475249

711 = 1977326743

712 = 13841287201

The units or ones digits have the pattern 1, 7, 9, 3, 1 , 7,  9, 3, ....  I extrapolate that the pattern will continue.  Now I must determine how the knowledge of this pattern will allow me to find the ones digit for 7189, a value much greater than 712 for example.

Since 70 results in the 1st number in the pattern, 7189 is the 190th number in the pattern.  The pattern repeats every fourth number.  How many multiples of four will we have in 190?  How many numbers beyond that multiple of 4 will we need to go?

There are 47 multiples of 4 in 190 with a remainder of 2 powers more.  47 * 4 = 188, so 190 falls on the second pattern number beyond the fourth number in the pattern, 7.

Every fourth number in the pattern has a 3 in the ones or units digit.  Two steps beyond this will always have a 7 in the pattern.

Extensions of the Problem

There is also a pattern at the 'ends' of my list values:  find 1, 7, 49, 43, 1, 7, 49, 43, ...  Is there a way to explain this pattern?

Author & Contact
Rita J. Rogers
rrogers@rockdale.k12.ga.us