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 N^{1}  N^{2}  N^{3}  N^{4}  N^{5}  N^{6}  N^{7}  N^{8}  N^{9} 
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 “ A^{n }” 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 “ A^{n }” 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 3^{2015}
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 4444^{2015}
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 4^{2012}
Solution: Here power value is even number. So unit digit of the given number is 6
Examples – 4 : Find the last digit of number 11^{123+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 19^{25}
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
Solution: First find unit place of 3^{993}
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.
= 3^{96 } here 96 multiple of 4 so last digit comes as 1
= ( – – – – 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
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
So unit digit of the given expression is 2
Related Topics :
Topics in Quantitative aptitude math for all types of exams
Shortcut Math Tricks for helpful to improve speed in all calculations
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