How to Find Unit Digit of a Power Number | Unit Digit Problems with Solutions

Finding Last Digit of any Number With Power | Unit place of a Number 

In Quantitative aptitude questions ask to find the last digit  and last two digits of  a power or large expressions. In this article explained different types of tools to serve as shortcuts to finding the last digits of an expanded power.

Find last digit of a number with power

First identify the pattern last digit (unit place) for power of numbers “N”

Digit N1 N2 N3 N4 N5 N6 N7 N8 N9
1 1 1 1 1 1 1 1 1
2 4 8 6 2 4 8 6 2
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

From the above table we can observe as follow

The last digit of power of 1, 5 & 6 is always comes same number as a unit place.

The last digit of power of 2 repeat in a cycle of numbers – 4, 8, 6 & 2

The last digit of power of 3 repeat in a cycle of numbers – 9, 7, 1 & 3

The last digit of power of 4 repeat in a cycle of numbers – 6 & 4

The last digit of power of 7 repeat in a cycle of numbers – 9, 3, 1, & 7

The last digit of power of 8 repeat in a cycle of numbers – 4, 2, 6 & 8

The last digit of power of 9 repeat in a cycle of numbers – 1 & 9

Explanation:

If Last digit ( Unit place ) of numbers having 1 , 5 & 6

  • ( – – – – 1)n =  ( – – – – 1)
  • (- – – – -5) n = ( – – – – 5)
  • (- – – – -6) n = (- – – – -6)

If the unit place ( Last digit ) of any number  “ An  having   2, 3, 7 or 8, then the unit place of  that number depends upon the value of  power “ n”  and follows

Expressed power “ n” Unit Place of ( – – -2)n Unit Place of ( – – -3)n Unit Place of ( – – -7)n Unit Place of ( – – -8)n
4x 6 1 1 6
4x + 1 2 3 7 8
4x + 2 4 9 9 4
4x + 3 8 7 3 2

If the unit place ( Last digit ) of any number  “ An  having    4 & 9  then the unit place of  that number depends upon the value of  power “ n”  and follows

Expressed power “ n” Unit Place of ( – – -4)n Unit Place of ( – – -9)n
2x (Even number) 6 1
2x + 1 (Odd number) 4 9

Last digit of a number questions

Examples  – 1 : Find last digit of the number  32015

Solution: The power 2015 can be written as [ (503 x 4) + 3 ]

So from the above table unit digit of given number is  – 7

Examples  – 2: Find  last digit of the number 44442015

Solution: Here power value is odd number

So last digit of the given number is 4

Hint: The last digit of any number having  “4”  then power having even number then unit place comes 6  and power having odd number then unit place comes 4

Example 3 : What is the last digit of the number 42012

Solution: Here power value is even number. So unit digit of the given number is 6

Examples  – 4 : Find the last digit of number 11123+5

Solution: Here The unit place having ” 1″ so the final number is also comes ” 1″ as a unit place

Examples  – 5 : Find the digit at the unit place of the number 1925

Hint: The last digit of any number having  “9”  then power having even number then unit place comes 1  and power having odd number then unit place comes 9

Solution:  Here power having odd number so final number unit place comes ” 9″

Examples  – 6: Find the digit at unit place of the number   3^{(99-3)^{50}}

Solution: First find unit place of  399-3

Hint:  Here the pattern of the last digits are 1 , 3, 9, 7, 1, 3 , 9 , 7 . . . . . . .   for the powers  4x , 4x+1 , 4x+2 , 4x+3 . . . . .  respectively.

= 396   here 96 multiple of 4 so last digit comes as 1

 3^{(99-3)^{50}}  = ( – – – – 1 )50 = ( – – – – – – – –  1)  i.e unit digit having 1 so final number unit place also comes 1

Find last digit of a large exponent

It is a remainder theorem application –  The last digit of an expression equals to remainder of that expression divided by 10.

Unit Digit problems with solutions

Examples  – 7: Find the unit digit of the expression 123 x 587 x 987 x 78

Solution: Here given expression 123 x 587 x 987 x 78 divided by 10 and find the remainder

\frac{123 \ \times \ 587 \ \times \ 987 \ \times\ 78}{10} = \frac{3 \ \times \ 7 \ \times \ 7 \ \times\ 8}{10}

= \frac{21 \ \times \ 56}{10} = \frac{6}{10} \overset{R}{\rightarrow} 6

So unit digit of the given expression is 6

Examples  – 8: Find the unit digit of the expression   578497 x 87548 x 25417

Solution: Here given expression 578497 x 87548 x 25417 divided by 10 and find the remainder

= 578497 x 87548 x 25417 / 10 = 7 x 8 x 7 / 10

= \frac{ 56 \ \times \ 7 }{10} = \frac{42}{10} \overset{R}{\rightarrow} 2

So unit digit of the given expression is 2

how to find last digit of any number with power | Find the last two digits of a large exponent | unit digit problems with solutions | unit digit concept

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Post Author: sivaalluri

My self Sivaramakrishna Alluri. Thank you for watching my blog friend

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