# Concept: Unit's Digit

**INTRODUCTION**

- Units's digit of sum of two numbers same as unit's digit of sum of their respective unit's digits.
- Units's digit of product of two numbers same as unit's digit of product of their respective unit's digits.

Example: Units digit of 126 + 589 = Units digit of 6 + 9 = 5

Example: Units digit of 126 × 589 = Units digit of 6 × 9 = 4

**CYCLICITY OF UNIT'S OF POWERS**

Let us calculate the unit's digit of various powers of 2.

Unit's digit of 2^{1} = **2**

Unit's digit of 2^{2} = **4**

Unit's digit of 2^{3} = **8**

Unit's digit of 2^{4} = **6**

Unit's digit of 2^{5} = **2**

Unit's digit of 2^{6} = **4**

Unit's digit of 2^{7} = **6**

Unit's digit of 2^{8} = **8**

Unit's digit of 2^{9} = **2**

...

∴ We see that the unit's digit of powers of 2 repeates after every 4 powers. Hence, we can say that unit's digit of powers of 2 has a cyclicity of 4 terms.

Unit's digit of 2^{4k} = **6**

Unit's digit of 2^{4k+1} = **2**

Unit's digit of 2^{4k+2} = **4**

Unit's digit of 2^{4k+3} = **8**

- 2 has a cyclicity of 4 terms.
- 3 has a cyclicity of 4 terms.
- 4 has a cyclicity of 2 terms.
- 5 has a cyclicity of 1 term.
- 6 has a cyclicity of 1 term.
- 7 has a cyclicity of 4 terms.
- 8 has a cyclicity of 4 terms.
- 9 has a cyclicity of 2 terms.
- 0 has a cyclicity of 1 term.

Unit's digit of powers of:

In general, we can say unit's digit of power of any number has a cyclicity of 4.

- Unit's digit of
- 4
^{odd number}=**4** - 4
^{even number}=**6** - Unit's digit of
- 9
^{odd number}=**9** - 9
^{even number}=**1** - Unit's digit of a
^{4k+n}= Unit's digit of a^{n}

where a is any single digit number. - Units's digit (...abc)
^{x}= Units's digit of c^{x}.

Example: Units digit of 126^{54}= Unit's digit of 6^{54}= 6 - Unit's digit of (5 × odd number) =
**5** - Unit's digit of (5 × even number) =
**0**

**Example**: Find unit's digit of (123)^{23}

**Solution:**

Unit's digit of (123)^{23} = Unit's digit of 3^{23}

= Unit's digit of 3^{4×5 + 3}

= Unit's digit of 3^{3}

= 7

**Example**: Find unit's digit of (23454)^{769}

**Solution:**

Unit's digit of (23454)^{769} = Unit's digit of 4^{769}

We know Unit's digit of 4^{odd number} = 4

∴ Unit's digit of (23454)^{769} = 4

**Example**: Find unit's digit of (582)^{945}

**Solution:**

Unit's digit of (582)^{945} = Unit's digit of 2^{945}

= Unit's digit of 2^{4×236 + 1}

= Unit's digit of 2^{1}

= 2

**Example**: Find unit's digit of ${\left(147\right)}^{{61}^{25}}$

**Solution:**

Unit's digit of ${\left(147\right)}^{{61}^{25}}$ = Unit's digit of ${\left(7\right)}^{{61}^{25}}$

Now, 61^{25} when divided by 4 leaves a remainder of 1 [Concept of Remainders]

∴ Unit's digit of ${\left(7\right)}^{{61}^{25}}$ = Unit's digit of 7^{4k + 1}

= Unit's digit of 7^{1}

= 7