Question

${4^{61}} + {4^{62}} + {4^{63}} + {4^{64}}$ is divisible by:
A. $3$
B. $10$
C. $11$
D. $13$

Answer 1

Hint: According to given in the question we have to find the number for which the given expression ${4^{61}} + {4^{62}} + {4^{63}} + {4^{64}}$ is divisible by. So, first of all we have to take the term or number which can be commended easily from the given expression.
Now, we have to solve the squares in the expression which is obtained after taking the number or term which can be taken.
Now, to solve or check that by which number the given expression can be divisible we have to rearrange the terms of the expression obtained after taking the common term and finding the square with the help of the formula as mentioned below:

Formula used:
$ \Rightarrow {a^n} = a.{a^{n - 1}}.............(A)$
Now, with the help of the formula (A) above we can determine the number from which the given expression can be divisible.

Complete step-by-step answer:
Step 1: First of all we have to take the term or number which can be commend easily from the given expression which is as mentioned in the solution hint and as we can see that from the given expression ${4^{61}}$can be taken as the common term from the given expression. Hence,
\[ = {4^{61}}(1 + {4^1} + {4^2} + {4^3})\]…………..(1)
Step 2: Now, we have to find the squares of the term to obtain the solution or find the number which can divide the given expression. Hence,
\[
   = {4^{61}}(1 + 4 + 16 + 64) \\
   = {4^{61}} \times 85 \\
 \]
Step 3: Now, we have to rearrange the terms of the expression as obtained in the solution step 3 which can be done with the help of the formula (A) which is as mentioned in the solution hint. Hence,
$
   = {4^{60}} \times 4 \times 85 \\
   = {4^{60}} \times 20 \times 17 \\
 $
Step 4: Now, as from the solution step 3 we can see that the number 20 is completely divisible by 3 hence, we can say that the given expression is divisible by 10.

Hence, with the help of the formula (A) we have determined that the given expression is divisible by 10. Therefore option (B) is correct.

Note:
It is necessary that we have to take the term or number common from the given expression which can be taken as a common term or number from the given expression. To check the divisibility for a number we have to rearrange the terms of the expression obtained after taking the common number or term as we know that ${a^n} = a.{a^{n - 1}}.$