## Finding Last Digit of any Number With Power | Unit place of a Number

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In Quantitative aptitude questions ask to find the last digit and last two digits of a power or large expressions. In this article explained different types of tools to serve as **shortcuts to finding the last digits** of an expanded power.

### Find last digit of a number with power

*First identify the pattern last digit (unit place) for power of numbers “N”*

Digit N^{1} |
N^{2} |
N^{3} |
N^{4} |
N^{5} |
N^{6} |
N^{7} |
N^{8} |
N^{9} |

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

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

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

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

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

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

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

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

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

**From the above table we can observe as follow**

The last digit of power of** 1, 5 & 6** is always comes same number as a unit place.

The last digit of power of** 2** repeat in a cycle of numbers –** 4, 8, 6 & 2**

The last digit of power of **3** repeat in a cycle of numbers – **9, 7, 1 & 3**

The last digit of power of** 4** repeat in a cycle of numbers – **6 & 4**

The last digit of power of **7** repeat in a cycle of numbers – **9, 3, 1, & 7**

The last digit of power of **8** repeat in a cycle of numbers –** 4, 2, 6 & 8**

The last digit of power of **9** repeat in a cycle of numbers – **1 & 9**

**Explanation:**

If Last digit ( Unit place ) of numbers having 1 , 5 & 6

- ( – – – – 1)
^{n}= ( – – – – 1) - (- – – – -5)
^{ n}= ( – – – – 5) - (- – – – -6)
^{ n}= (- – – – -6)

If the unit place ( Last digit ) of any number **“ A ^{n }”** having

**2, 3, 7 or 8**, then the unit place of that number depends upon the value of power

**“ n”**and follows

Expressed power “ n” |
Unit Place of ( – – -2)^{n} |
Unit Place of ( – – -3)^{n} |
Unit Place of ( – – -7)^{n} |
Unit Place of ( – – -8)^{n} |

4x |
6 | 1 | 1 | 6 |

4x + 1 |
2 | 3 | 7 | 8 |

4x + 2 |
4 | 9 | 9 | 4 |

4x + 3 |
8 | 7 | 3 | 2 |

If the unit place ( Last digit ) of any number ** “ A ^{n }”** having

**4 & 9**then the unit place of that number depends upon the value of power

**“ n”**and follows

Expressed power “ n” |
Unit Place of ( – – -4)^{n} |
Unit Place of ( – – -9)^{n} |

2x (Even number) |
6 | 1 |

2x + 1 (Odd number) |
4 | 9 |

#### Last digit of a number questions

**Examples – 1 :** Find last digit of the number 3^{2015}

**Solution:** The power 2015 can be written as [ (503 x 4) + 3 ]

So from the above table unit digit of given number is – 7

**Examples – 2:** Find last digit of the number 4444^{2015}

**Solution:** Here power value is odd number

So last digit of the given number is 4

Hint: The last digit of any number having “4” then power having even number then unit place comes 6 and power having odd number then unit place comes 4

**Example 3 :** What is the last digit of the number 4^{2012}

**Solution:** Here power value is even number. So unit digit of the given number is 6

**Examples – 4 :** Find the last digit of number 11^{123+5}

**Solution:** Here The unit place having ” 1″ so the final number is also comes ” 1″ as a unit place

**Examples – 5 :** Find the digit at the unit place of the number 19^{25}

Hint: The last digit of any number having “9” then power having even number then unit place comes 1 and power having odd number then unit place comes 9

Solution: Here power having odd number so final number unit place comes ” 9″

**Examples – 6:** Find the digit at unit place of the number

**Solution:** First find unit place of 3^{99-3}

Hint: Here the pattern of the last digits are 1 , 3, 9, 7, 1, 3 , 9 , 7 . . . . . . . for the powers 4x , 4x+1 , 4x+2 , 4x+3 . . . . . respectively.

= 3^{96 } here 96 multiple of 4 so last digit comes as 1

= ( – – – – 1 )^{50} = ( – – – – – – – – 1) i.e unit digit having 1 so final number unit place also comes 1

### Find last digit of a large exponent

It is a** remainder theorem application** – The last digit of an expression equals to remainder of that expression divided by 10.

#### Unit Digit problems with solutions

**Examples – 7**: Find the unit digit of the expression** 123 x 587 x 987 x 78**

Solution: Here given expression **123 x 587 x 987 x 78** divided by 10 and find the remainder

So unit digit of the given expression is 6

**Examples – 8: **Find the unit digit of the expression **578497 x 87548 x 25417**

Solution: Here given expression **578497 x 87548 x 25417** divided by 10 and find the remainder

= 578497 x 87548 x 25417 / 10 = 7 x 8 x 7 / 10

So unit digit of the given expression is 2

**Related Topics :**

Topics in Quantitative aptitude math for all types of exams

Shortcut Math Tricks for helpful to improve speed in all calculations

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