Units digit:  Units digit of a number is the digit in the one’s place of the number. It is the rightmost digit of the number.

For example,

• The unit digit of the 6847 is 7
• The unit digit of the number 265 is 5.

In Quantitative Aptitude questions ask to find the last digit and last two digits of power or large expressions. We will study different types of tricks to serve as shortcuts to finding the last digits of an expanded power.

Finding the unit digit of the sum of two numbers (Addition)

This is easy and we can find simply adding the unit digits of the two numbers.

For Example,  What is the units digit of \(2,067+24,908\)? Hopefully, you see that adding only the units digits gives us 5(8+7=15, 5 is the unit digit with a Carry 1) without completing the addition.

Finding the unit digit of the product of two numbers (Multiplication)

Finding the units digit of a product is also easy. We only need to multiply the units digits of the two numbers.

For example, the units digit of \(1,024 \cdot 21,097\) is 8 (7 x 4 = 28, the unit digit is 8 in this case).

Finding the unit digit of perfect squares

The units digit of a perfect square is simple as well, using the same technique as above.
For Example, the units digit of \(947,866^2\) is 6 (Squaring only the 6 gives us the units digit 6).

Find the last digit of a number with power

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

\(
\begin{array}{|c|c|c|c|c|c|c|c|c|}
\hline \text { Digit } \mathrm{N}^1 & \mathrm{~N}^2 & \mathrm{~N}^3 & \mathrm{~N}^4 & \mathrm{~N}^5 & \mathrm{~N}^6 & \mathrm{~N}^7 & \mathrm{~N}^8 & \mathrm{~N}^9 \\
\hline 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\
\hline 2 & 4 & 8 & 6 & 2 & 4 & 8 & 6 & 2 \\
\hline 3 & 9 & 7 & 1 & 3 & 9 & 7 & 1 & 3 \\
\hline 4 & 6 & 4 & 6 & 4 & 6 & 4 & 6 & 4 \\
\hline 5 & 5 & 5 & 5 & 5 & 5 & 5 & 5 & 5 \\
\hline 6 & 6 & 6 & 6 & 6 & 6 & 6 & 6 & 6 \\
\hline 7 & 9 & 3 & 1 & 7 & 9 & 3 & 1 & 7 \\
\hline 8 & 4 & 2 & 6 & 8 & 4 & 2 & 6 & 8 \\
\hline 9 & 1 & 9 & 1 & 9 & 1 & 9 & 1 & 9 \\
\hline
\end{array}
\)

From the above table, we can observe as follow
The last digit of power of \(1,5 \& 6\) is always comes the same number as a unit place.
The last digit of power of 2 repeats in a cycle of numbers \(-4,8,6 \& 2\)
The last digit of power of 3 repeats in a cycle of numbers \(-9,7,1 \& 3\)
The last digit of power of 4 repeats in a cycle of numbers \(-6 \& 4\)
The last digit of power of 7 repeats in a cycle of numbers \(-9,3,1, \& 7\)
The last digit of power of 8 repeats in a cycle of numbers \(-4,2,6 \& 8\)
The last digit of the power of 9 repeats in a cycle of numbers \(-1 \& 9\)

Important Points to Remember

Trick No.1:

If the Last digit ( Unit place ) of the number has \(0,1,5 \& 6\)

• \((\cdots0)^n=(\cdots0)\)
• \((\cdots1)^n=(\cdots1)\)
• \((\cdots5)^{\mathrm{n}}=(\cdots5)\)
• \((\cdots6)^n=(\cdots6)\)

Example 1: Find the unit digit of the following numbers;

(a) \(185^{563}\)

(b) \(271^{6987}\)

(c) \(156^{25369}\)

(d) \(190^{654789321}\)

Solution:

(a) 5 is the answer as the number is in the form of \(5^{{n}}\).

(b) 1 is the answer as the number is in the form of \(1^{{n}}\).

(c) 6 is the answer as the number is in the form of \(6^{{n}}\).

(d) 0 is the answer as the number is in the form of \(0^{{n}}\).

Trick No.2:

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 as shown in the table below.

\(
\begin{array}{|c|c|c|c|c|}
\hline \text { Expressed power ” } n \text { ” } & \text { Unit Place of }(—2)^n & \text { Unit Place of }(—3)^n & \text { Unit Place of }(—7)^n & \text { Unit Place of }(—8)^n \\
\hline 4 x & 6 & 1 & 1 & 6 \\
\hline 4 x+1 & 2 & 3 & 7 & 8 \\
\hline 4 x+2 & 4 & 9 & 9 & 4 \\
\hline 4 x+3 & 8 & 7 & 3 & 2 \\
\hline
\end{array}
\)

 

Example 2: Find the last digit of the number \(3^{2015}\)

Solution:

The power 2015 can be written as [ \((503 \times 4)+3\) ] which is in the form of \(4x+3\).
So from the above table unit digit of the given number is \(7\).

Example 3: \(\text { Find the Unit digit of } 287^{562581}\)

Solution:

The power 562581 can be written as [ \((140,645 \times 4)+1\) ] which is in the form of \(4x+1\). Since the last digit of the given number is 7 and from the above table, the unit digit of the given number is 7.

Trick No.3:

If the unit place ( Last digit ) of any number ” \(A\) “” having \(4 \& 9\) then the unit place of that number depends upon the value of power ” \(\mathbf{n}\) ” and follows as shown in table below.

\(
\begin{array}{|c|c|c|}
\hline \text { Expressed power ” } n \text { ” } & \text { Unit Place of }(–4)^n & \text { Unit Place of }(—9)^n \\
\hline 2 x \text { (Even number) } & 6 & 1 \\
\hline 2 x+1 \text { (Odd number) } & 4 & 9 \\
\hline
\end{array}
\)

 

Example 4: Find last digit of the number \(4444^{2015}\)

Solution: Here power value is an odd number. So the last digit of the given number is 4 as per the table above.

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 5: What is the last digit of the number \(4^{2012}\)

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

Example 6: Find the last digit of number \(11^{123+5}\)

Solution:

Since the unit place is having ” 1 “, the final number also comes ” 1 ” as a unit place.

Example 7: Find the digit at the unit place of the number \(19^{25}\)

Solution:

Since the power has an odd number so the final number of the unit place comes to 9. 

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

Example 8: Find the digit at unit place of the number \(3^{(99-3)^{50}}\)

Solution:

First find unit place of \(3^{99-3}\)
The number \(3^{96}\), here 96 is a multiple of 4 so the last digit comes as 1.
\(3^{(99-3)^{50}}=(—1)^{50}=\left(—–1\right)\) i.e unit digit having 1, so the final number unit place also comes 1.

Example 9: Find the unit digit of the expression \(123 \times 587 \times 987 \times 78\)

Solution:

Here given expression \(123 \times 587 \times 987 \times 78\) divided by 10 and find the remainder
\(
\frac{123 \times 587 \times 987 \times 78}{10}=\frac{3 \times 7 \times 7 \times 8}{10}
\)
\(
=\frac{21 \times 56}{10}=117 \stackrel{R}{\rightarrow} 6
\)
So unit digit of the given expression is 6.