## Patterns in the Last Digit of Powers of Integers

Where a number is raised to the power of some other number a pattern emerges in the last digit. The following table illustrates all last dits 1 - 9 and powers up to 9.

Power | |||||||||

Last Digit | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |

0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

2 | 2 | 4 | 8 | 6 | 2 | 4 | 8 | 6 | 2 |

3 | 3 | 9 | 7 | 1 | 3 | 9 | 7 | 1 | 3 |

4 | 4 | 6 | 4 | 6 | 4 | 6 | 4 | 6 | 4 |

5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 |

6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 |

7 | 7 | 9 | 3 | 1 | 7 | 9 | 3 | 1 | 7 |

8 | 8 | 4 | 2 | 6 | 8 | 4 | 2 | 6 | 8 |

9 | 9 | 1 | 9 | 1 | 9 | 1 | 9 | 1 | 9 |

If the last digit is 0 or 5, the last digit of any power is 0 or 5 respectively.

If the last digit is 1 or 6, the last digit of any power is 1 or 6 respectively.

If the last digit is 2 or 7, the last digit repeats every time the power increases by 4.

If the last digit is 3 or 8, the last digit repeats every time the power increases by 4.

If the last digit is 4 or 9 the last digit repeats every time the power increaes by 2.

The behaviour of the last digit is the same if the last digit increases by 5.

If the power is 1, 3, 5, 7, 9 all the integers 0 - 9 appear as last digits.

If the power is 2 0r 6 only the numbers 0, 1, 4, 5, 6 or 9 may appear as last digits.

If the power is 4 0r 8 only the numbers 0, 1, 5 or 6 may appear as last digits.