Question

What is the remainder when \[{2^{63}}\] is divided by 7?
a. 2
b. 4
c. 1
d. 5

Answer 1

Hint: Here in this question we have to find the remainder of \[{2^{63}}\] when it is divided by 7. The number is in the form of exponential form. First we simplify the exponential number and by the concept of combination topic and obtain the required result for the given question.

Complete step by step solution:
The exponential number is defined as the number that is multiplied by the number is itself. The power of 2 is very large, so we can’t multiply the number that many times. So we use the concept of binomial theorem that makes it easy to simplify.
Now consider the given exponential number \[{2^{63}}\]
This exponential number can be written as \[{2^{3(21)}}\]
When the number 2 is multiplied thrice the product is 8.
Therefore the given exponential number is written as \[{8^{21}}\].
The number 8 is written as 7 + 1. Therefore the exponential number is written as \[{(7 + 1)^{21}}\]
We solve this by using the binomial theorem.
The binomial theorem is given by \[{(a + b)^n} = {}^n{C_0}{a^n}{b^0} + {}^n{C_1}{a^{n - 1}}{b^1} + {}^n{C_2}{a^{n - 2}}{b^2} + ... + {}^n{C_n}{a^0}{b^n}\]
Therefore \[{(7 + 1)^{21}}\] is written as \[{(7 + 1)^{21}} = {}^{21}{C_0}{7^{21}}{1^0} + {}^{21}{C_1}{7^{20}}{1^1} + {}^{21}{C_2}{7^{19}}{1^2} + ... + {}^{21}{C_{21}}{7^0}{1^{21}}\]
On simplifying we can write it as
\[ \Rightarrow {(7 + 1)^{21}} = {}^{21}{C_0}{7^{21}} + {}^{21}{C_1}{7^{20}} + {}^{21}{C_2}{7^{19}} + ... + {}^{21}{C_{21}}\]
Take 7 as common
\[ \Rightarrow {(7 + 1)^{21}} = 7({}^{21}{C_0}{7^{20}} + {}^{21}{C_1}{7^{19}} + {}^{21}{C_2}{7^{18}} + ... + {}^{21}{C_{20}}) + {}^{21}{C_{21}}\]
On further simplifying we get
\[ \Rightarrow {(7 + 1)^{21}} = 7({7^{20}} + {}^{21}{C_1}{7^{19}} + {}^{21}{C_2}{7^{18}} + ... + {}^{21}{C_{20}}) + 1\]
The terms in the braces is the multiple of 7, so when we divide it by 7 the remainder will be 0. But we have 1 last term.
Therefore the remainder when \[{2^{63}}\] is divided by 7 is 1.

So, the correct answer is Option C.

Note: When number is in the form of exponential form. When the power of an exponential number is very large, we use the concept of binomial theorem. By using the binomial theorem we can simplify the term in a very easy manner. We should know about the tables of multiplication.