Find Unit digit of an expression

To understand the concept of unit digit, we must know the concept of cyclicity . This concept is mainly about the unit digit of a number and its repetitive pattern on being divided by a certain number

The concept of unit digit can be learned by figuring out the unit digits of all the single digit numbers from 0 - 9 when raised to certain powers.
These numbers can be broadly classified into three categories for this purpose:

1. Digits 0, 1, 5 & 6:

When we observe the behaviour of these digits, they all have the same unit's digit as the number itself when raised to any power, i.e. 0^n = 0, 1^n =1, 5^n = 5, 6^n = 6. Let's apply this concept to the following example.
Example: Find the unit digit of following numbers:
185⁵⁶³
Answer= 5
271⁶⁹⁸⁷
Answer= 1
156²⁵³⁶⁹
Answer= 6
190^654789321
Answer= 0

2. Digits 4 & 9:

Both these numbers have a cyclicity of only two different digits as their unit's digit.
Let us take a look at how the powers of 4 operate: 4¹ = 4,
4² = 16,
4³ = 64, and so on.
Hence, the power cycle of 4 contains only 2 numbers 4 & 6, which appear in case of odd and even powers respectively. 
Likewise, the powers of 9 operate as follows:
9¹ = 9,
9² = 81,
9³ = 729, and so on.
Hence, the power cycle of 9 also contains only 2 numbers 9 & 1, which appear in case of odd and even powers respectively.
So, broadly these can be remembered in even and odd only, i.e. 4 odd = 4 and 4 even = 6. Likewise, 9 odd = 9 and 9 even = 1.

Example: Find the unit digit of following numbers:
189^562589743
Answer = 9 (since power is odd)
27969^8745832
Answer = 1(since power is even)
154^258741369
Answer = 4 (since power is odd)
194^65478932
Answer = 6 (since power is even)

3. Digits 2, 3, 7 & 8:

These numbers have a power cycle of 4 different numbers.
2¹ = 2, 2² = 4, 2³ = 8 & 2⁴ = 16 and after that it starts repeating.
So, the cyclicity of 2 has 4 different numbers 2, 4, 8, 6.
3¹ = 3, 3² = 9, 3³ = 27 & 3⁴ = 81 and after that it starts repeating.
So, the cyclicity of 3 has 4 different numbers 3, 9, 7, 1.
7 and 8 follow similar logic.
So these four digits i.e. 2, 3, 7 and 8 have a unit digit cyclicity of four steps.

Example 1: Find the Unit digit of 287⁵⁶²⁵⁸¹
Solution Step 1: We know that the cyclicity of 7 is 4.
Step 2: Divide the power 562581 by 4.
By doing that, we get a remainder=1.
Step 3: 1st power in the power cycle of 7 is 7.
Hence, the answer is 7.

Example 2: Find the Unit digit of 13445 * 54336
Solution: Cyclicity of 5 & 6 is 1. Since 5*6=30, the unit digit of given expression is 0.
 

Key Learning:

• You must remember the power cycles of all the digits from 1-10.
• If the power cycle of number has 4 different digits, divide the power by 4, find the remaining power and calculate the unit’s digit using that. Similarly, if the power cycle of number has 2 different digits, divide the power by 2, find the remaining power and calculate the unit’s digit using that