**Unit Digit of a Number with Power :**

In this section, you will learn how to find unit digit of a number in the form a^{b}.

To identify the unit digit of a number with some power, we must be aware of cyclicity.

Cyclicity of any number is about the last digit and how they appear in a certain defined manner.

**Example 1 :**

Let us consider the values of 2^{n}, where n = 1, 2, 3, ...........

2^{1 } = 2

2^{2} = 4

2^{3} = 8

2^{4} = 16

2^{5} = 32

2^{6} = 64

In the above calculations of 2^{n},

We get unit digit 2 in the result of 2^{n}, when n = 1.

Again we get 2 in the unit digit of 2^{n}, when n = 5.

That is, in the fifth term.

So, the cyclicity of 2 is 4.

**Example 2 :**

Let us consider the values of 3^{n}, where n = 1, 2, 3, ...........

3^{1} = 3

3^{2} = 9

3^{3} = 27

3^{4} = 81

3^{5} = 243

3^{6} = 729

In the above calculations of 3^{n},

We get unit digit 3 in the result of 3^{n}, when n = 1.

Again we get 3 in the unit digit of 3^{n}, when n = 5.

That is, in the fifth term.

So, the cyclicity of 3 is 4.

In the same way, we can get cyclicity of others numbers as shown in the table below.

Number 1 2 3 4 5 6 7 8 9 10 |
Cyclicity of a number 1 4 4 2 1 1 4 4 2 1 |

**Example 1 :**

Find the unit digit of 32^{24}.

**Step 1 : **

Take the unit digit in 32 and find its cyclicity.

The unit digit of 32 is '2' and its cyclicity is 4.

**Step 2 : **

Divide the exponent 24 by the cyclicity 4.

**Step 3 :**

When 24 is divided by 4, the remainder is zero.

Because the remainder is zero, we can get the unit digit of 2^{8}, from the last value of the cyclicity of 2^{n}_{.}

The last value of the cyclicity of 2^{n} is

2^{4} = 16

The unit digit of 16 is '6'.

Therefore, the unit digit of 32^{24 }is 6.

In step 3 of the above example, what if the remainder is not zero?

It has been explained in the next example.

**Example 2 :**

Find the unit digit of 32^{27}.

**Step 1 : **

Take the unit digit in 32 and find its cyclicity.

The unit digit of 32 is '2' and its cyclicity is 4.

**Step 2 : **

Divide the exponent 27 by the cyclicity 4.

**Step 3 :**

When 27 is divided by 4, the remainder is 3.

Take the remainder 3 as power of 2 (unit digit of 32).

2^{3} = 8

Therefore, the unit digit of 32^{27 }is 8.

**Question 1 :**

Find the unit digit of (3547)^{153}_{.}

**Solution :**

In (3547)^{153}, unit digit is 7.

The cyclicity of 7 is 4. Dividing 153 by 4, we get 1 as remainder.

7^{1} = 7

Therefore, the unit digit of (3547)^{153} is 7.

**Question 2 :**

Find the unit digit of (264)^{102}_{.}

**Solution :**

In (264)^{102}, unit digit is 4.

The cyclicity of 4 is 2. Dividing 102 by 2, we get 0 as remainder.

Since the remainder is 0, unit digit will be the last digit of a cyclicity number.

4^{2} = 16 (the unit digit is 6)

Unit digit of 4^{102} is 6.

Therefore, the unit digit of (264)^{102} is 6.

**Question 3 :**

What is the unit digit in the product

(7)^{105}

**Solution :**

The unit digit is 7.

The cyclicity of 7 is 4. Dividing 105 by 4, we get 1 as remainder.

So,

7^{1} = 7

The unit digit is 7.

Therefore the unit digit of (7)^{105} is 7.

**Question 4 :**

Find unit digit of (3^{65} x 6^{59} x 7^{71}).

**Solution :**

Cyclicity of 3 is 4. Dividing 65 by 4, we get the remainder 1.

Then,

3^{1} = 3

So, the unit digit of 3^{65 }is 3.

Cyclicity of 6 is 1. Dividing 59 by 1, we get the remainder 0.

Then,

6^{1} = 6

So, the unit digit of 6^{59 }is 6.

Cyclicity of 7 is 4. Dividing 71 by 4, we get the remainder 3..

Then,

7^{3} = 343

The unit digit of 343 is '3'.

So, the unit digit of 7^{71 }is also 3.

Product of unit digits :

= 3 x 6 x 3

= 54

The unit digit of the product is 4.

Therefore, the unit digit of (3^{65} x 6^{59} x 7^{71}) is 4.

After having gone through the stuff given above, we hope that the students would understood how to find the unit digit of a number.

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