Answer

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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.**

**Additional Information:**

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 \[2\].

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