Unit Digit Calculation Notes (How to Calculate Unit Digit within seconds) | UPSC Prelims Paper 2 CSAT - Quant, Verbal & Decision Making

Shortcut Trick for Finding the Units Digits of Large Powers

Easy Way of finding the Units Digit of Large Powers | How to find the Units Digit of Large Powers| Trick to find the Unit Digits of Large Powers.

In Exam, you may find few questions based on finding the Units Digits of Large Powers. A typical example of such questions is listed below:

(a) Find the Units Place in  (785)⁹⁸ + (342)³³ + (986)⁶⁷

(b) What will come in Units Place in  (983)⁸⁵ -  (235)³⁷

These questions can be time consuming for those students who are unaware of the fact that there is a shortcut method for solving such questions. Don't worry if you don't know the shortcut already because we are providing it today.
 

Finding the Unit Digit of Powers of 2

1. First of all, divide the Power of 2 by 4.
2. If you get any remainder, put it as the power of 2 and get the result using the below given table.
3. If you don't get any remainder after dividing the power of 2 by 4, your answer will be (2)⁴ which always give 6 as the remainder

Let's solve few Examples to make things clear.

(1) Find the Units Digit in (2)³³
Sol -
Step-1:: Divide the power of 2 by 4. It means, divide 33 by 4.
Step-2: You get remainder 1.
Step-3: Since you have got 1 as a  remainder , put it as a power of 2 i.e (2)¹.
Step-4: Have a look on table, (2)¹=2. So, Answer will be 2


(2) Find the Unit Digit in (2)⁴⁰
Sol -
Step-1:: Divide the power of 2 by 4. It means, divide 40 by 4.
Step-2: It's completely divisible by 4. It means, the remainder is 0.
Step-3: Since you have got nothing as a  remainder , put 4 as a power of 2 i.e (2)⁴.
Step-4: Have a look on table, (2)⁴=6. So, Answer will be 6

 

Finding the Unit Digit of Powers of 3 (same approach)

1. First of all, divide the Power of 3 by 4.
2. If you get any remainder, put it as the power of 3 and get the result using the below given table.
3. If you don't get any remainder after dividing the power of 3 by 4, your answer will be (3)⁴ which always give 1 as the remainder

Let's solve few Examples to make things clear.

(1) Find the Units Digit in (3)³³
Sol -
Step-1:: Divide the power of 3 by 4. It means, divide 33 by 4.
Step-2: You get remainder 1.
Step-3: Since you have got 1 as a  remainder , put it as a power of 3 i.e (3)¹.
Step-4: Have a look on table, (3)¹=3. So, Answer will be 3


(2) Find the Unit Digit in (3)³²
Sol -
Step-1:: Divide the power of 3 by 4. It means, divide 32 by 4.
Step-2: It's completely divisible by 4. It means, the remainder is 0.
Step-3: Since you have got nothing as a  remainder , put 4 as a power of 3 i.e (3)⁴.
Step-4: Have a look on table, (3)⁴=1. So, Answer will be 1
 

Finding the Unit Digit of Powers of 0,1,5,6

The unit digit of 0,1,5,6 always remains same i.e 0,1,5,6 respectively for every power.

Finding the Unit Digit of Powers of 4 & 9

In case of  4 & 9, if powers are Even, the result will be 6 & 4. However, when their powers are Odd, the result will be 1 & 9. The same is depicted below.

• If the Power of 4 is Even, the result will be 6
• If the Power of 4 is Odd, the result will be 4
• If the Power of 9 is Even, the result will be  1
• If the Power of 9 is Odd, the result will be 9.

For Example - 

• (9)⁸⁴ = 1
• (9)²¹ = 9
• (4)⁶⁴ = 6
• (4)⁶³ = 4

Finding the Unit Digit of Powers of 7 (same approach)

1. First of all, divide the Power of 7 by 4.
2. If you get any remainder, put it as the power of 7 and get the result using the below given table.
3. If you don't get any remainder after dividing the power of 7 by 4, your answer will be (7)⁴ which always give 1 as the remainder

Let's solve few Examples to make things clear.

(1) Find the Units Digit in (7)³⁴
Sol -
Step-1:: Divide the power of 7 by 4. It means, divide 34 by 4.
Step-2: You get remainder 2.
Step-3: Since you have got 2 as a  remainder , put it as a power of 7 i.e (7)².
Step-4: Have a look on table, (7)²=9. So, Answer will be 9


(2) Find the Unit Digit in (7)⁸⁴
Sol -
Step-1:: Divide the power of 7 by 4. It means, divide 84 by 4.
Step-2: It's completely divisible by 4. It means, the remainder is 0.
Step-3: Since you have got nothing as a  remainder , put 4 as a power of 7 i.e (7)⁴.
Step-4: Have a look on table, (7)⁴=1. So, Answer will be 1

Finding the Unit Digit of Powers of 8 (same approach)

1. First of all, divide the Power of 8 by 4.
2. If you get any remainder, put it as the power of 8 and get the result using the below given table.
3. If you don't get any remainder after dividing the power of 8 by 4, your answer will be (8)⁴ which always give 6 as the remainder

Let's solve few Examples to make things clear.

(1) Find the Units Digit in (8)³⁴
Sol -
Step-1:: Divide the power of 8 by 4. It means, divide 34 by 4.
Step-2: You get remainder 2.
Step-3: Since you have got 2 as a  remainder , put it as a power of 8 i.e (8)².
Step-4: Have a look on table, (8)²=4. So, Answer will be 4


(2) Find the Unit Digit in (8)³²
Sol -
Step-1:: Divide the power of 8 by 4. It means, divide 32 by 4.
Step-2: It's completely divisible by 4. It means, the remainder is 0.
Step-3: Since you have got nothing as a  remainder , put 4 as a power of 8 i.e (8)⁴.
Step-4: Have a look on table, (8)⁴=1. So, Answer will be 6

Now, you can easily solve questions based on finding the Unit's Digit of large powers. Lets try at least a few.

(a) Find the Units Place in  (785)⁹⁸ + (342)³³ + (986)⁶⁷

Sol : 5 + 2 + 6 = 13 . So answer will be 3 .


(a) Find the Units Place in  (983)⁸⁵ -  (235)³⁷

Sol :  3 - 5 = 13 - 5 = 8  . So answer will be 8 . In this question, we have considered 3 as 13 because 3-5= -2 which is negative which is not possible.