Finding unit digit

Search TakshilaOnline.com
In competitive exams, you often come across questions like find unit digit of 53343. Obviously you can't solve such a big calculation manually. Lets learn how to solve these kind of questions quickly and accurately.

Before we move to some interesting and easy to digest concepts, lets try some easy questions and build the concept slowly.

Question 1: Find the unit digit of 105.
105 = 10 X 10 X 10 X 10 X 10  = 100000; so the unit digit of 105 = 0

Question 2: Find the unit digit of 303.
303 = 30 X 30 X 30 = 27000; so the unit digit of 303 = 0

Question 3: Find the unit digit of 315.
315 = 31 X 31 X 31 X 31 X 31 = 28629151; so the unit digit of 315 = 1

Question 4: Find the unit digit of 1013.
1013 = 101 X 101 X 101 = 1030301; so the unit digit of 1013 = 1

Question 5: Find the unit digit of 35^5.
355 = 35 X 35 X 35 X 35 X 35 = 52521875; so the unit digit of 355 = 5

Question 6: Find the unit digit of 75^3.
753 = 75 X 75 X 75 = 28629151; so the unit digit of 753 = 5

Learning 1:
If the unit digit of a number is 0, 1 , 5 then no matter how many times you multiply this number with itself, unit digit of result would always be 0, 1, 5 respectively.

That means we could have done above six questions without doing a single calculation.
                                                                ***********************

But what to do if unit digit is not 0, 1 or 5?

Question 7: Find the unit digit of 35.
Solution: 
You can simply multiply 3, five times.
35= 3 X 3 X 3 X 3 X 3 = 243
So the unit digit will be 3.

Question 8: Find the unit digit of 135.
Solution: 
How will you approach it? You may do –
13 X 13 X 13 X 13 X 13 = 371293
So the unit digit will be 3.

Learning 2:

Did you notice unit digit of 35 is same as unit digit of 135?

So the unit digit of a number depends only on the digit which is at unit place. Rest other digits do not impact unit digit of result.

That’s means unit digit of 37 or 237 or 45737 will be same.

So next time when you are asked to get unit digit 43^5 or 13423^5, you will simply ignore the digits  which are on the left side of unit digit i.e. in this case you will ignore all the numbers written left of 3.

So you can conclude that:
Unit digit of 37 = Unit digit of 23^7 = Unit digit of 4573^7

So now how to calculate unit digit of 37:
37= 3 X 3 X 3 X 3 X 3 X 3 X3 = 2187

So unit digit of 37 or 237 or 45737 = 7 

                                                                ***********************

Question 9: Calculate unit digit of 17 X 89
17 X 89= 10413 so the required unit digit =3

Question 10: Calculate unit digit of 7 X 9
7 X 9= 63 so the required unit digit =3

Learning 3:
To calculate unit digit of multiplication of two number a and b, we only consider unit digits of a and b.
Unit digit of a X b = Unit digit of a X Unit digit of b

Question 11: Find the unit digit of 233 X 312
Unit digit of 233 X 312= Unit digit of 233 X Unit digit of 312
                                   = 3 X 2 = 6

                                                                ***********************

Question 12: Find the unit digit of 237234.

So far you have learnt that 
Unit digit of 237234 = Unit digit of 7234

i.e. Unit digit of 237234 = 7 X 7 X 7 X 7……………. 234 times

Puzzled? How to move further? Obviously it is not practically possible to complete this calculation manually.
So what’s the way?

                                                                ***********************

Suppose you are asked to find out unit digit of a number x^y where x and y can be any positive integer.

You can approach these kind of questions in following way:

Number given = xy

First of all check if y is multiple of 4 or not. i.e. if you divide y by 4, it leaves any remainder or not.

Case 1: When y is NOT multiple of 4

So if y is not multiple of 4, that means if you divide y by 4, there must be some remainder.

That means I can represent y in following format:
y = 4 q + r
where r is remainder.

If you divide any number by 4, possible remainders could be 1, 2 or 3 only i.e. 0 < r < 4

Now let’s calculate unit digit of x^y

The unit digit of x^y = unit digit of x^r

Lets take one example to simplify the things.

Question 13: Find unit digit of 7^33
Solution: If I compare given number with x^y, then x = 7, y = 33

Is y divisible by 4?

33 = 4 X 8 + 1
So the remainder r = 1

We already know that:

unit digit of xy = unit digit of xr

unit digit of 733 = unit digit of 71 =7

So the required unit digit is 7. This way you can avoid the cumbersome calculation of 7 X 7 X 7 X…… 33 times. 

Sounds easy? Lets try a different scenario now.

                                                                ***********************

Case 2: When y is multiple of 4

Again there are two possibilities:

Case 2 (i): If the unit digit of x = 2, 4, 6, 8 (even number)
Case 2 (ii): If the unit digit of x = 1, 3, 7, 9 (odd number except 5)

Let’s discuss each case ony by one.
                                                                ***********************
Case 2 (i): If the unit digit of x = 2, 4, 6, 8 (even number)

Then unit digit of x^y = 6

Question 14: Find unit digit of 2^20.

Solution: x=2, y=20

Is y (=20) divisible by 4 = yes
Unit digit of x =2 

So unit digit of 2^20 must be 6.

By the way 2^20 = 1048576
                                                                ***********************

Case 2 (ii): If the unit digit of x = 1, 3, 7, 9 (odd number except 5)

Then unit digit of x^y = 1

Question 15: Find unit digit of 4344.

Solution: x = 43, y = 44

Is y (=44) divisible by 4 = Yes

So unit digit of 4344 = 1

OR you can simply consider: unit digit of 43^44 = unit digit of 3^44 (we have already discussed it above)

x = 3, y = 44

Is y (=44) divisible by 4 = Yes

So unit digit of 4344 = unit digit of 344 = 1

                                                                ***********************
Lets summarize what we have discussed so far.

You have to find out unit digit of x^y

1. Notice if unit digit of x is 0, 1 or 5 if yes then irrespective of value of y, unit digit of x^y will be 0, 1 or 5 respectively.

2. If unit digit of x is NOT 0, 1 or 5.
Then
Check if y is divisible by 4.

        CASE 1: if y is not divisible by 4

        Then unit digit of x^y = unit digit of x^r where r is remainder when y is divided by 4.

        CASE 2: if y is divisible by 4
 
                  CASE 2(i) if unit digit of x is 2, 4, 6, 8 unit digit of xy must be 6
                  
                  CASE 2(ii) if unit digit of x is 3, 7, 9 unit digit of xy must be 1


                                                                ***********************


Now you try following questions and discuss in comments section of this page.

Question: Find unit digit of 2828 - 2424


Question: Find unit digit of 4343 - 2222

Question: The unit digit in the expression 343X643X743 is.......  (SSC-CGL Tier-2 2015)
a. 6
b. 7
c. 3
d. 4

More detailed article and questions on this topic are coming soon. I will explain how exactly we are concluding the unit digits.

Practice various multiple choice questions here.

Happy Learning!

Small appreciation please!


       
   



  Posted on Friday, October 23rd, 2015 at 10:15 AM under   Reasoning