Find the unit digit of $\left( {{7^{95}} - {3^{58}}} \right)$
A) 6
B) 4
C) 3
D) None of these
Answer
279.9k+ views
Hint: To find the unit digit of $\left( {{7^{95}} - {3^{58}}} \right)$, first of all let it be equal to N. Now, break the power of 7 and 3 separately in simple terms so we can find their unit digits. After breaking their powers, we will get their unit digits and now we just need to subtract them to get our answer.
Complete step by step solution:
In this question, we are given an expression and we need to find the unit digit of that expression.
Given expression: $\left( {{7^{95}} - {3^{58}}} \right)$
Now, let this expression be equal to N. Therefore, we get
$ \Rightarrow N = \left( {{7^{95}} - {3^{58}}} \right)$
Now, as the power is very large, we cannot directly find the unit digit of these numbers. So, we need to break the power of these numbers in simple powers.
Here, we know that ${7^4} = 2401$. Therefore, the unit digit of ${7^4}$ will be equal to 1.
$
\Rightarrow {7^4} = 2401 \equiv 1 \\
\Rightarrow {\left( {{7^4}} \right)^{23}} \equiv {1^{23}} \\
\Rightarrow {7^{92}} \equiv 1 \\
$
Hence, the unit digit of ${7^{92}}$ will be equal to 1.
Now, we need to find the unit digit of ${7^{95}}$. Therefore, we multiply ${7^{92}}$ with ${7^3}$.
${7^3} = 343 \equiv 3$
$
\Rightarrow {7^{92}} \cdot {7^3} \equiv 1 \cdot 3 \\
\Rightarrow {7^{95}} \equiv 3 \\
$
Hence, the unit digit of ${7^{95}}$ will be equal to 3.
Now, for ${3^{58}}$, we know that ${3^4} = 81$ and so the unit digit for ${3^4}$ will be equal to 1.
$
\Rightarrow {3^4} \equiv 1 \\
\Rightarrow {\left( {{3^4}} \right)^{14}} \equiv {\left( 1 \right)^{14}} \\
\Rightarrow {3^{56}} \equiv 1 \\
$
But we need to find the unit digit of ${3^{58}}$, so we multiply ${3^{56}}$ with ${3^2}$.
${3^2} = 9 \equiv 9$
$
\Rightarrow {3^{56}} \cdot {3^2} \equiv 1 \cdot 9 \\
\Rightarrow {3^{58}} \equiv 9 \\
$
Hence, the unit digit of ${3^{58}}$ will be equal to 9. Therefore,
$ \Rightarrow N = 3 - 9$
Now, we cannot subtract 9 from 3, so we take 1 carry from the number at the tens place. So, the equation becomes
$ \Rightarrow N = 13 - 9 = 4$
Therefore, the unit digit for $\left( {{7^{95}} - {3^{58}}} \right)$ will be equal to 4.
Hence, option (B) is the correct answer.
Note:
Note that finding the unit digit with such a high power is not possible directly. We need to break the powers into simple powers.
Here, note that we have to break the power in such a way that we get the unit digit as 1 only because anything raised to 1 will be 1 only. For example: We took $ {3^4} = 81 \equiv 1$.
Complete step by step solution:
In this question, we are given an expression and we need to find the unit digit of that expression.
Given expression: $\left( {{7^{95}} - {3^{58}}} \right)$
Now, let this expression be equal to N. Therefore, we get
$ \Rightarrow N = \left( {{7^{95}} - {3^{58}}} \right)$
Now, as the power is very large, we cannot directly find the unit digit of these numbers. So, we need to break the power of these numbers in simple powers.
Here, we know that ${7^4} = 2401$. Therefore, the unit digit of ${7^4}$ will be equal to 1.
$
\Rightarrow {7^4} = 2401 \equiv 1 \\
\Rightarrow {\left( {{7^4}} \right)^{23}} \equiv {1^{23}} \\
\Rightarrow {7^{92}} \equiv 1 \\
$
Hence, the unit digit of ${7^{92}}$ will be equal to 1.
Now, we need to find the unit digit of ${7^{95}}$. Therefore, we multiply ${7^{92}}$ with ${7^3}$.
${7^3} = 343 \equiv 3$
$
\Rightarrow {7^{92}} \cdot {7^3} \equiv 1 \cdot 3 \\
\Rightarrow {7^{95}} \equiv 3 \\
$
Hence, the unit digit of ${7^{95}}$ will be equal to 3.
Now, for ${3^{58}}$, we know that ${3^4} = 81$ and so the unit digit for ${3^4}$ will be equal to 1.
$
\Rightarrow {3^4} \equiv 1 \\
\Rightarrow {\left( {{3^4}} \right)^{14}} \equiv {\left( 1 \right)^{14}} \\
\Rightarrow {3^{56}} \equiv 1 \\
$
But we need to find the unit digit of ${3^{58}}$, so we multiply ${3^{56}}$ with ${3^2}$.
${3^2} = 9 \equiv 9$
$
\Rightarrow {3^{56}} \cdot {3^2} \equiv 1 \cdot 9 \\
\Rightarrow {3^{58}} \equiv 9 \\
$
Hence, the unit digit of ${3^{58}}$ will be equal to 9. Therefore,
$ \Rightarrow N = 3 - 9$
Now, we cannot subtract 9 from 3, so we take 1 carry from the number at the tens place. So, the equation becomes
$ \Rightarrow N = 13 - 9 = 4$
Therefore, the unit digit for $\left( {{7^{95}} - {3^{58}}} \right)$ will be equal to 4.
Hence, option (B) is the correct answer.
Note:
Note that finding the unit digit with such a high power is not possible directly. We need to break the powers into simple powers.
Here, note that we have to break the power in such a way that we get the unit digit as 1 only because anything raised to 1 will be 1 only. For example: We took $ {3^4} = 81 \equiv 1$.
Recently Updated Pages
Define absolute refractive index of a medium

Find out what do the algal bloom and redtides sign class 10 biology CBSE

Prove that the function fleft x right xn is continuous class 12 maths CBSE

Find the values of other five trigonometric functions class 10 maths CBSE

Find the values of other five trigonometric ratios class 10 maths CBSE

Find the values of other five trigonometric functions class 10 maths CBSE

Trending doubts
The ray passing through the of the lens is not deviated class 10 physics CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Difference Between Plant Cell and Animal Cell

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

What is pollution? How many types of pollution? Define it

What is the nlx method How is it useful class 11 chemistry CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

What is the difference between anaerobic aerobic respiration class 10 biology CBSE
