How to Find Unit Digit?

He e’s a uestio fo ou!

Find the unit digit in the product of 567 x 693 x 391 x 453 x 188

While finding the unit digit a typical mind will first multiply all the numbers and will take the

unit digit out of the final number.

Now, as we need to find out the unit digit, our focus must be on the unit digit only. In the

same question, we just need to multiply the unit digits (leaving tens digit every time we get a

dou le digit u e ) a d e’ll get the a s e .

Let’s see.

Now for the last few years unit digit questions are being asked in large powers of a number.

Ex. Find the unit digit in 13137

The answer is 3.

Before I explain this, kindly have a look at the patter how the powers of different digits

behave.

Steps to be taken:

Multiplying the unit digits 567 x 693 x 391 x 453 x 188

We get, 7 x 3 = 21 (leaving the tens digit 2)

1 x 1 = 1

1 x 3 = 3

3 x 8 = 24

The answer will be 4.

Now, for 4 and 9 we have a similar patter that is as follows:

Now, different powers of 2 and 8 carry the same pattern. Let’s see hat that is.

For n power of 0, we always get 0 as the unit digit. And same is the case with 1, 5 and 6.

It means,

0n

= 0, 1n

= 1, 5n

= 5, 6n

= 6

For 0 as power of any number N we always get the unit digit as 1. It means,

N0

= 1

For every odd power of 4 and 9 the unit digit is 4 and 9 respectively and for every even

power of 4 and 9 the unit digit is 6 and 1 respectively.

It means,

41 = 4

42 = 16

Similarly, for the power of 9

91 = 9

92 = 81

Unit digit in 21

= 2 & Unit digit in 81

= 8

Unit digit in 22

= 4 & Unit digit in 82

= 64 = 4

Unit digit in 23

= 8 & Unit digit in 83

= 512 = 2

Unit digit in 24

= 16 = 6 & Unit digit in 84

= 4096 = 6

Unit digit in 25

= 32 = 2 & Unit digit in 85

= 32768 = 8

Unit digit in 26

= 64 = 4 & Unit digit in 86

= 262144 = 4

Unit digit in 27

= 128 = 8 & Unit digit in 87

= 2097152 = 2

Unit digit in 28

= 256 = 6 & Unit digit in 88

= 16777216 = 6

.

From the above table we can see that the le e uals .

When the exponent is a multiple of 4, the unit digit is 6 in both the cases of base 2 or 8.

For example 24 has unit digit 6, 2

8 has unit digit 6.

Similarly, 84 has unit digit 6, 8

8 has unit digit 6.

Ex: What is the unit digit in 13521354

?

Solution:

As the unit digit in the base value is 2. We can re-write the number as 21354

To find the unit digit with base 2 we first need to convert the power in the multiples of 4

and for that we just need to divide the last two digits by 4.

(Divisibility Test of 4: To check the divisibility of a number by 4, the last two digits of the

number must be either 00 or a multiple of 4.)

In the given power the last two digits are 54. When we divide 54 by 4 we get the

remainder power as 2.

So our question shrinks to 22

that is very easy to find which is 4. The answer is 4.

Ex: What is unit digit in 178719

?

Solution:

As the unit digit in the base value is 8. We can re-write the number as 8719

To find the unit digit with base 8 we first need to convert the power in the multiples of 4

as we did in the case of base 2.

In the given power the last two digits are 19. When we divide 19 by 4 we get the

remainder power as 3.

The question hence shrinks to 83

the unit digit in which is 2. The answer, therefore, is 2.

Now we are left with the last two digit which are 3 and 7. These two digits too share a patter

which is similar in itself.

Let’s ha e a look.

Unit digit in 31

= 3 & Unit digit in 71

= 7

Unit digit in 32

= 9 & Unit digit in 72

= 49 = 9

Unit digit in 33

= 27 = 7 & Unit digit in 73

= 343 = 3

Unit digit in 34

= 81 = 1 & Unit digit in 74

= 2401 = 1

Unit digit in 35

= 243 = 3 & Unit digit in 75

= 16807 = 7

Unit digit in 36

= 729 = 9 & Unit digit in 76

= 117649 = 9

Unit digit in 37

= 2187 = 7 & Unit digit in 77

= 823543 = 3

Unit digit in 38

= 6561 = 1 & Unit digit in 78

= 5764801 = 1

.

.

From the above table we can see that the le e uals .

When the exponent is a multiple of 4, the unit digit in both the cases is 1.

For rest of the powers we can use the remainder power method.

For example 34 has unit digit 1, 3

8 has unit digit 1.

Similarly, 74 has unit digit 1, 7

8 has unit digit 1.

For rest of the powers we can find the unit digit with the remainder powers after

converting the maximum of powers into the multiples of 4.

Ex. Find the unit digit in 3193321

Solution:

Let’s fi st sho te the uestio e-writing it as 3321

Now, dividing the last two digits of the power that is 21 by 4 we get the remainder 1.

(The powers which get converted into multiples of 4 will give unit digit 1. So, we neglect

that.)

The question hence shrinks itself to 31 that is 3. The answer, therefore, is 3.

Some Mixed Example:

Ex. 1. Find the unit digit in 3178

x 9613

x 7783

Solution:

Taki g o e ase digit at a ti e, let’s fi st fi d out the u it digit i 178

Using the method explained above, we get the remainder power 2 when we divide the

last two digits of the power by 4.

So, the unit digit in 3178

= Unit digit in 378

= Unit digit in 32

= 9

Unit digit in 9613

= 9An odd power

= 9

Unit digit in 7783

= Unit digit in 783

= Unit digit in 73

= Unit digit in 343 = 3

Now, we are left with three unit digits that go like:

9 x 9 x 3 = 3 (as we are concerned only with the unit digit)

The final unit digit is 3.

Ex. 2. Find the unit digit in 1331677 − 77714

Solution:

Let’s fi st fi d the u it digit i di iduall .

Unit digit in 1331677

= Unit digit in 31677

= Unit digit in 377

= Unit digit in 31

= 3

Unit digit in 77714

= Unit digit in 7714

= Unit digit in 714

= Unit digit in 72

= 9

The final equatio e get is, − 9

(Now, what you generally do when you subtract a bigger digit from a smaller one? You

take a a digit f o the te s digit. He e, too e’ll do the sa e as he e a e t i g to fi d the u it digit, it does ’t ea that the te s o hu d eds digit does ’t e it.)

The final unit digit hence will be = 4

The answer is 4.