Title
Units of Seven
Problem Statement
Find the units digit of 7^{189}. Determine a general rule for finding
the units digit of 7^{n},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 oncomputer calculator for as a
computational tool.
Investigation/Exploration of the
Problem
7^{0} = 1
7^{1} = 7
7^{2} = 49
7^{3 }= 343
7^{4 }= 2401
7^{5 }= 16807
7^{6 }= 117649
7^{7 }= 823543
7^{8 }= 5764801
7^{9 }= 40353607
7^{10 }= 282475249
7^{11 }= 1977326743
7^{12} = 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 7^{189}, a value much greater than 7^{12
}for example.
Since 7^{0} results in the
1st number in the pattern, 7^{189} 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
