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# The remainder when ${2^{100}}$ is divided by $7$ isA) 1B) 2C) 3D) 4

Last updated date: 11th Aug 2024
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Hint:
Here, we will convert the power of 2 to its simplest form by using the exponential property. Then we will use the division algorithm to divide the simplified form and find the remainder from there.

Formula Used:
Exponential formula ${a^{mn}} = {\left( {{a^m}} \right)^n}$

Complete step by step solution:
Here, we have to find the remainder when ${2^{100}}$ is divided by $7$.
Now we will rewrite ${2^{100}}$ in the powers of $10$ by using the exponential formula ${a^{mn}} = {\left( {{a^m}} \right)^n}$ , we get
$\dfrac{{{2^{100}}}}{7} = \dfrac{{{{\left( {{2^{10}}} \right)}^{10}}}}{7}$
Thus, the remainder when ${2^{100}}$ is divided by $7$ is equal to the remainder when ${2^{10}}$ is divided by $7$. So, we get
$\Rightarrow \dfrac{{{2^{10}}}}{7} = \dfrac{{1024}}{7}$
Now we know that the product of 146 and 7 is 1022 which is 2 less than 1024.
So, we can say that 146 is the quotient, 2 is the remainder.
By using the division algorithm, we get
$1024 = \left( {146} \right)7 + 2$
Thus, the remainder when ${2^{10}}$ is divided by $7$ is $2$ .
Therefore, the remainder when ${2^{100}}$ is divided by $7$ is $2$.

Thus, option (B) is the correct answer.

We know that Division algorithm states that for any integer $a$ and $b$ be any positive integer such that there exists two unique integers, then we have $a = qb + r$ where $r$ is an integer equal to $0$ or less than $b$ i.e., $0 \le r < b$. We say $a$ is the dividend, $b$ is the divisor, $q$ is the quotient, $r$ is the remainder. Euler’s theorem states that if $n$ is relatively prime to $m$ then ${m^{\phi \left( n \right)}}$ divided by $n$ gives 1 as the remainder i.e., ${\rm{Remainder}}\left( {\dfrac{{{m^{\phi \left( n \right)}}}}{n}} \right) = 1$

Note:
We will find the remainder by using Euler’s theorem. We can also find the remainder when ${2^{100}}$ is divided by $7$ by using Euler’s theorem
$\dfrac{{{2^{100}}}}{7} = \dfrac{{2 \cdot {{\left( {{2^3}} \right)}^{33}}}}{7}$
By simplifying, we get
$\Rightarrow \dfrac{{{2^{100}}}}{7} = \dfrac{{2 \cdot {{\left( 8 \right)}^{33}}}}{7}$
By dividing the terms, we get we get
$\Rightarrow \dfrac{{{2^{100}}}}{7} = 2 \cdot {\left( 1 \right)^{33}}$
By simplifying, we get
$\Rightarrow \dfrac{{{2^{100}}}}{7} = 2 \cdot 1$
By multiplying, we get
$\Rightarrow \dfrac{{{2^{100}}}}{7} = 2$
Therefore, the remainder when ${2^{100}}$ is divided by $7$ is $2$.