**How to find unit digit of a number raised to power**

To understand the concept unit digit, try to understand the concept of cyclicity and the approach to find the unit digit of a number when the number is of the form .

Unit digit of = unit digit of

Now consider the following cases :

**Case :1 **When z= 0, 1, 6 or 5 , the unit digit of is=0,1, 5 or 6 respectively.

Example : (i)Find unit digit of

Sol. unit digit of = unit digit of = 5 ( any power of 5 gives unit digit 5)

(ii) Find unit digit of

Sol. Unit digit of = unit digit of

So unit digit of will be equal to 6.

**Case II: **When z is equal 4 or 9.

Let us take power of 4 :

power of 9 :

We can easily see that cycle repeats after 2. So the cyclicity of 4 and 9 is 2. It also follows from the above pattern that when the power of 4 is odd , the unit digit of = 4 and when the power of 4 is even then unit digit of = 6.

Similarly for 9 also , when power of 9 is odd , unit digit of =9 and when power of 9 is even , unit digit of .

Example : Find the unit digit of .

Sol. Unit digit of = unit digit of = 1 ( power of 9 is even)

**Case 3: When z is 2, 3, 7 or 8. **

Clearly cycle of 2,3,7 and 8 repeats after 4. So the cyclicity of 2,3,7 and 8 is **4. **

When z = 2,3,4,7,8 and 9 , following table is useful in finding the last digit of .

value of z | then divide ‘n’ | If remainder is | then the last digit of |

4 and 9 | by 2 (cyclicity of 4 and 9 is 2) | 1 | = unit digit of ( ) |

0 | = unit digit of ( ) | ||

2, 3, 7 and 8 | by 4 (cyclicity of 2 ,3,7 and 8 is 4) | 1 | =unit digit of ( ) |

2 | =unit digit of ( ) | ||

3 | =unit digit of ( ) | ||

0 | =unit digit of ( ) |

**Problems based on unit digit of a number raised to some power**

**Example 1. Find the unit digit of . **

Sol. The unit digit 287 is 7 so divide 562581 by 4 . On dividing 562581 by 4 we get remainder 1.

Thus the unit digit of = unit digit of = unit digit of =** 7 **

**Example 2. Find the unit digit of **.

Sol. Unit digit of = unit digit of

= unit digit of =**7** ( On dividing 263 by 4 we get remainder as 3)

Unit digit of = unit digit of

Therefore unit digit of = unit digit of (7 x 1) = 7. ** Ans. 7**

**Example 3 : Find unit digit of .**

Sol . Unit digit of = unit digit of = 1 ( power of 9 is even)

Unit digit of = unit digit of = unit digit of =7 ( when divided by 4 gives remainder 3)

Therefore unit digit of = 1-7 =-6 =-6+10=4 ( when unit digit comes negative then add 10)

Ans. **4**

**Example 4: Find unit digit of .**

Sol. Since every n! where n >3 , is divisible by 4. So when 11! is divided by 4 gives remainder 0

Therefore unit digit of = unit digit of = unit digit of 16 =**6**.

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