Endings

1
Endings

• A Maths Investigation by
• Rhiannon McKay
• and
• Marie Formosa

2
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?

3
Example of working method.
4
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!

5
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 1525
• 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.

6
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

7
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.

8
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

9
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

10
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

11
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.