## Patterns in the Last Digit of Powers of Integers

| 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.