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					Endings
A Maths Investigation by
   Rhiannon McKay
          and
    Marie Formosa
               Aims
 Investigate the end digits of
   integers raised to powers.

What do you notice for single end digits?

What do you notice for double end digits?
 Example of working method.
Number   ^2         ^3          ^4         ^5           ^6             ^7

  1            1           1          1            1              1               1

  2            4           8         16           32             64             128

  3            9          27         81          243            729            2187

  4           16          64         256        1024           4096           16384

  5           25         125         625        3125          15625           78125

  6           36         216     1296           7776          46656          279936

  7           49         343     2401           16807        117649          823543

  8           64         512     4096           32768        262144         2097152

  9           81         729     6561           59049        531441         4782969

  10          100        1000    10000      100000           1000000        10000000
          Single End Digits
       Single end digits have many patterns
Eg: 2, 12 and 22 raised to powers all end in a
 repeated pattern of 2, 4, 8, 6.

Prediction: Numbers which end with 2 will all
            end in the same pattern.
Result:      TRUE!
Do any powers produce single or double digits
 equal to the base? Do any powers produce
            them for all the bases?
   5 raised to powers end with 25 same as 15&25
   6 raised to powers all end in 6, until 6 ^ 20.
   11 raised to powers end with 1. The double end
    digits add 10 with each power.
   Integers ending with 9 raised to powers alternate
    between 1 and 9. Double end digits ending in 1 go
    subtract by 20. Double end digits ending in 9 add
    20.
   21- same as 11, but adds 20 instead of 10.
      Methods, Trial & Error
Find a method for finding the single base to
  any power…
 Trial = All numbers ending with 6
         (6, 16, 26 etc) will end in 6.
  Error = 6 ^ 20 ends with 0
            16 ^13 ends with 0
            26 ^ 11 end with 0
          Error Conclusion…
 As the integer and power increase, the end number
  will be 0. The first time this occurs is after 24 ^
  11.
 Numbers that follow this are:
  18 ^12 ; 15 ^ 13 ; 12 ^14 – integer subtracts 3
  when the power is increased by 1.
~ Prediction: the next integer raised to a power with
  an end number of 0 will be 9 ^ 15.
~ Result: WRONG! It is 11 ^ 15… there is no quick
  method of finding which numbers will end with 0.
  Single end digits of bases that
           equal to 10
Odds- When it is two separate end digits raised to an
  odd power their end digit will also add to 10.
Eg: 2 ^ 3 = 8
                     =10
   8 ^ 3 =512
Evens- When it is two separate end digits raised to
  an even power their end digit will be the same.
Eg: 4 ^ 2 =16
                   = same
   6 ^ 2 =36
    Double end digits of bases that
            equal to 100
 A similar thing happens for double end digits equal to 100
Odds- the last 2 digits equal to 100
Eg:    23 ^ 5 =      6436343
                                     = 100
       77 ^ 5 = 2706784157

Evens- the last 2 digits are the same
Eg:    72 ^ 2 = 5184
                                        = same
       28 ^ 2 = 784
            Double end digits
 There is no easy way to find double end digit patterns.
 Patterns we did find!
-5,25,45,65 etc all end in 25
-15,35,55,75 etc have an ending pattern of 25,75…
-For 7 the pattern is 47,49,30,10…
-For 9 numbers ending in 9 go up by 20. Numbers ending in
  1 go down by 20.
-For those integers that end in 1 the double end digit
  increases by the tens number.
   Eg: 21= 21,41,61,81,01,21…
         Endings Conclusion
There are many patterns in integers raised to powers
  however none can be obtained easily.

Patterns do emerge but they disappear as the power and
  integer increase.

				
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posted:4/1/2011
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