Question

The remainder when \[{2^{100}}\] is divided by \[7\] is
A) 1
B) 2
C) 3
D) 4

Answer 1

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 \[2\].