Units digit of an expression 1-1

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Units digit: ConceptUnits digit: Exercise

1. Find the units digit of $23 \times 47 \times 84$
a. 4
b. 6
c. 8
d. 0

Answer: A

Explanation:
We have to consider only units digits. $2\boxed3 \times 4\boxed7 \times 8\boxed4$
$ \Rightarrow 3 \times 7 \times 4$
$ \Rightarrow 2\boxed1 \times 4$ ($\because$ consider only units digit of any product.)
$ \Rightarrow 1 \times 4 = 4$

2. Find the units digit of ${2^4} + {4^2} + {5^6} + {6^5}$
a. 2
b. 4
c. 6
d. 8

Answer: A

Explanation:
${2^4} + {4^2} + 5 + 6$ $= 16 + 16 + 5 + 6 = \boxed3$
($\because$ any power of 5 or 6 gives same units digit.)

3. Find the units digit of $3^{20}$
a. 1
b. 3
c. 9
d. 7

Answer: A

Explanation:
As cyclicity for 3 is 4, we have to divide the power by 4.
$\begin{array}{rl}
3^{20} &= {\left( {{3^4}} \right)^5} \\[4px]
& = {\left( 1 \right)^5} \\[4px]
& = 1
\end{array}$

4. Find the units digit of ${2^{25}}$
a. 2
b. 4
c. 6
d. 8

Answer: A

Explanation:
$\begin{array}{rl}
{2^{25}} &= {\left( {{2^4}} \right)^k}{.2^1} \\[4px]
& = {\left( 6 \right)^k}{.2^1} \\[4px]
& = 6 \times 2 \\[4px]
& = 2
\end{array}$

5. Find the units digit of ${2^{75}} \times {3^{50}} \times {7^{100}}$
a. 2
b. 4
c. 6
d. 8

Answer: A

Explanation:

$\begin{array}{rl}
{2^{75}} \times {3^{50}} \times {7^{100}} &= {\left( {{2^4}} \right)^k}{.2^3} \times {\left( {{3^4}} \right)^m}{.3^2} \times {\left( {{7^4}} \right)^n} \\[4px]
& = {\left( 6 \right)^k}{.2^3} \times {\left( 1 \right)^m}{.3^2} \times {\left( 1 \right)^n} \\[4px]
& = (6 \times 8) \times 9 \times 1 \\[4px]
& = 8 \times 9 \times 1\\[4px]
& = 2
\end{array}$

Note: Here we don't have to find the values of $k,\,m,\,n$ as the powers of 6 and 1 gives same units digit. So we focus only on the remainder.

6. Find the units digit of ${4^{75}} \times {9^{50}}$
a. 4
b. 9
c. 6
d. 8

Answer: A

Explanation:
$\text{Units digit of}\,4^k =
\begin{cases}
4, & \text{if k is odd} \\
6, & \text{if k is even}
\end{cases}$
$\text{Units digit of}\,9^k =
\begin{cases}
9, & \text{if k is odd} \\
1, & \text{if k is even}
\end{cases}$
$\therefore$ ${4^{75}} \times {9^{50}}$ $=4 \times 1 = 4$

Number System: BasicsNumber System: HCF and LCMNumber System: Factors and CoprimesNumber System: Divisbility Rules Number System: Power of a number in a Factorial Number System: Units digit of an expression Number System: Last two digits of an expressionNumber System: Level 2 and 3 ExerciseNumber System: Base SystemNumber System: Last non zero digit of a factorial (LNZ)Number System: Last two non zero digits of a factorialNumber System: High level Exercise