# endings by xiangpeng

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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.
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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