FINDING THE LAST DIGIT OF ANY NUMBER TO ANY POWER

FINDING THE LAST DIGIT OF ANY NUMBER TO ANY POWER (CYCLICITY)

H₂OH₂H₂3²¹O

Email: debasisbasak1997@gmail.com

Before we go into the topic, let’s see the power of all the numbers from 1 to 9.

Note that we can find some pattern in their units digit after a fixed cycle..

NOTE: -- is used wherever the rest of the number is not of concern

CYCLE -- It is the units digit cycle.

N   N²   N³   N⁴   N⁵        CYCLE

1    1     1     1     1          1 1 1 1 

2    4     8    16   32         2 4 6 8

3     9  --7   --1   --3          9 7 1 3 

4   --6  --4   --6   --4          6 4 6 4

5    -5   -5    -5   --5          5 5 5 5

6    -6   -6   -6    --6          6 6 6 6

7   --9  --3  --1    --7          9 3 1 7 

8   --4  --2 --6     --8          4 2 6 8

9   --1  --9 --1     --9          1 9 1 9

     N       N²     N³     N⁴    N⁵    CYCLE

Here we see that in all of the numbers from 1 to 9 the pattern in powers repeats after every cycle. This is what we can use to find the last digit of any number to any power. 

Lets start with an easy example to explain the basic principles and concept behind what is done.

1) Suppose we to find the last digit of 733³⁴

STEP 1

If we think about this question we can see that in 733 we are only concerned with the units digit so the tens hundreds and so on digits have no role in our working. 

Thus 475450453 or 342343 or 3 does not change our answer because the units digit of any power is only dependent on the units digit of the number we are taking .so now the

question reduces to-

What is the last digit of 3³⁴ ?

STEP 2

We have seen in the table above that the pattern of 3 is 3, 9, 7 , 1 so it repeats after every cycle of 4 suppose the question was thus in a more mathematical form we can

write

3^(4k+1) =  ......3

3^(4k+2) =  ......9

3^(4k+3) =  ......7

3^(4k+4) =  ......1

This means that if we find the remainder of the power from 4 we can find the last digit.

Now since,

3³⁴ =  3² = ......9

this matches out form 3^(4k+1) 

3³⁴ => 3² = 9

So the last digit of the Question is 9.

2) Find last digit of 4324836⁴²³¹⁴²³⁴

Just by looking (Vilokanam Sutra) we can say that the units digit is same as

6⁴²³¹⁴²³⁴

pattern of 6 is all 6’s so we can simply say units digit is 6. 

3) Find the units digit of 4377⁵⁷⁹⁷

STEP 1

Just like before –

7⁵⁷⁹⁷

STEP 2

Find remainder of 5797 from 4 we get 1

Answer is same as 7¹ =7 

Units Digit is 7

4)Units digit of 523⁵²⁴5

S1(Step 1) - 

3⁵²⁴5

S2 - 5245 ÷4 leaves Rem 1

3¹ = 3

Units digit is 3.

5) Units Digit of 4234⁴⁷⁶³²

S1 - 4

47632

S2 – Rem of 47632 from 4 = 0

Don’t mistake this as 40 because this is wrong!

Instead we write 44 because it matches the type 3^(4k+4) = --6

So our answer is 6 not 1. 

6) Units digit of 524567²⁵⁴⁵³⁴⁵⁵⁴³⁵⁴³⁵⁴⁵

(I have purposely taken a big power to show how easy it can be!!)

S1 – As we did in the last Questions we write it as

524567²⁵⁴⁵³⁴⁵⁵⁴³⁵⁴³⁵⁴⁵

S2 - In the last question I had told to find the remainder from 4. But finding remainder from 4 in this example is no easy task. Here is where osculation comes handy . I am not going into osculation because Swami Bharati Krishna Tirthaji Maharaja has already done a 

wonderful and elaborate work on it in his book (See reference).

RULE– ANY number divided by 4 will leave same remainder as the last two digits divided by 4.

Here is what i am saying-

2545345543543545 leaves some remainder when divided by 4 .

However this remainder is same as that of 45 when divided by 4 

which can easily be found to be 1.

So, 7¹ = 7

UD (Units digit) is 7.

7) UD of 84732423059⁹⁸⁰⁸⁹⁰⁷⁷⁸⁹⁰⁸⁸ 

Don’t be afraid of big numbers because they don't mean anything to us!!

S1- 9⁹⁸⁰⁸⁹⁰⁷⁷⁸⁹⁰⁸⁸

S2- 9808907789088 when divided by 4 leaves same Rem as 88 div by 4.

9⁴ => --1

UD is 1.

8) Find 40234986783⁴²³⁴⁴³⁵³⁴⁵²³⁴⁵ unit digit.

Easy, or Hard? You decide.

S1- 3⁴²³⁴⁴³⁵³⁴⁵²³⁴⁵

S2- 3⁴⁵ = 3¹

UD is 3

SUMMARY

(This is all you need to remember. The tables and figures were to convey the concepts)

* Any number to any power is no problem. We are just concerned with the last digit of the number and last two digits of the power 

RULE– ANY number divided by 4 will leave same remainder as the last two digits divided by 4.

* If the power is divisible by 4 we replace the power by 4.

For example

 3²⁴ =  3⁴

 7²⁸ = 7⁴

Here are some questions (As an extra Challenge Try doing these mentally)

1) 7637³⁴² mod 10                          [9]

2) 5639⁴²⁹⁸ mod 10                        [1]

3) 8363⁴⁵³⁵ mod 10                        [7]

4) 7854594⁴⁵²³⁴⁵⁴ mod 10             [6]

5) 56782⁹³⁵⁵⁴³⁵⁰ mod 10