Concept of identifying the unit digit

For the concept of identifying the unit digit, we have to first familiarize with the concept of cyclicity. Cyclicity of any number is about the last digit and how they appear in a certain defined manner. Let’s take an example to clear this thing:

Let us see the cyclicity chart of 2 is.
2¹ =2
2² =4
2³ =8
2⁴=16
2⁵=32

Let us have a close look the order above. We see that as 2 is multiplied every-time with its own self, the last digit changes. On the 4^th multiplication, 2⁵ has the same unit digit as 2¹. This shows us the cyclicity of 2 is 4 that is after every fourth multiplication, the unit digit will be two.

Cyclicity table:

The cyclicity table for numbers is given as below:

Now let us use the concept of cyclicity to calculate the Unit Digit of a number

Example 1.

What is the unit digit of the expression 4⁹⁹³?

Now we have two methods to solve this but we choose the best way to solve it i.e. through cyclicity
We know the cyclicity of 4 is 2 from the cyclic table.

Let us have a look:

4¹ =4, 4² =16, 4³ =64, 4⁴ =256

From above it is clear that the cyclicity of 4 is 2.  Now with the cyclicity number i.e. with 2 divide the given power i.e. 993 by 2 what will be the remainder the remainder will be 1 so the answer when 4 raised to the power one is 4.So the unit digit in this case is 4.

We can try many more examples, let us think of any number like this and let us calculate its unit digit and then check it with the help of a calculator.

Note : If the remainder becomes zero in any case then the unit digit will be the last digit of  a^{cyclicity number} 

Where a is the given number and cyclicity number is shown in above figure.

Example 2

The digit in the unit place of the number 7²⁹⁵ X 3¹⁵⁸ is

A.  7
B.  2
C.  6
D.  4

Solution

The Cyclicity table for 7 is as follows:
7¹ =7, 7² =49, 7³ = 343, 7⁴ = 2401

Let’s divide 295 by 4 and the remainder is 3. Thus, the last digit of 7²⁹⁵ is equal to the last digit of 7³ i.e. 3.
The Cyclicity table for 3 is as follows:
3¹ =3, 3² =9, 3³ = 27, 3⁴ = 81, 3⁵ = 243

Let’s divide 158 by 4, the remainder is 2. Hence the last digit will be 9.
Therefore, unit’s digit of (7²⁹⁵ X 3¹⁵⁸) is unit’s digit of product of digit at unit’s place of 7⁹²⁵ and 3¹⁵⁸

= 3  9 = 27. Hence option (A) is the answer.