Question

For every positive integer $n$, prove that ${7^n} - {3^n}$ is divisible by 4.

Answer 1

Before attempting this question one should have prior knowledge about the method of mathematical induction and the steps which are included in the method of mathematical induction also remember that if the last 2 digits of a number are a multiple of, then it is a true divisible of 4, use this information to approach the solution of the problem.

Complete step-by-step solution:
As per the question, we need to find out the ${7^n} - {3^n}$ is divisible by 4.
To show that the $P\left( n \right) = {7^n} - {3^n}$ is divisible by 4 let use the method of mathematical induction.
The steps are as follows
STEP I: In this step we have to show that the for n = 1 P (1) is true
let n = 1
substituting the value of n in the equation ${7^n} - {3^n}$ we get
$\Rightarrow {7^1} - {3^1}$ = 4 …………....…...(equation 1)
since we know that 4 is divisible by 4 therefore
for P (1)
This case is true
STEP II: Show that the for p (m) is true
Assume P (n) is true for n = m
Then ${7^m} - {3^m}$ is divisible by 4
Therefore ${7^m} - {3^m} = 4k$
$\Rightarrow$ ${7^m} = {3^m} + 4k$; $k \in n$ (equation 2)
Therefore, for case P (m) is true
STEP III: Show that also P (m + 1) is true
Therefore n = m + 1

\Rightarrow {7^{m + 1}} - {3^{m + 1}} \\\ \Rightarrow {7.7^m} - {3^{m + 1}} \\\ \end{gathered} $$ Substituting the value form equation 2 we get $$\begin{gathered} \Rightarrow {7^1}.\left( {{3^m} + 4k} \right) - {3^{m + 1}} \\\ \Rightarrow {7^1}{.3^m} - {3^{m + 1}} + 28k \\\ \Rightarrow {7^1}{.3^m} - {3^m}{.3^1} + 28k \\\ \Rightarrow {3^m}\left( {{7^1} - {3^1}} \right) + 28k \\\ \end{gathered} $$ Now substituting the value form the equation 1 $$\begin{gathered} \Rightarrow {3^m}\left( 4 \right) + 28k \\\ \Rightarrow {3^m}\left( 4 \right) + 28k \\\ \end{gathered} $$ Now taking $${3^k} + 7k$$ as k’ $$\begin{gathered} \Rightarrow 4\left[ {{3^k} + 7k} \right] \\\ \Rightarrow 4k' \\\ \end{gathered} $$ $ \Rightarrow P\left( {m + 1} \right)$is true **I, II, AND III induction say P (n) is true for all positive integers n.** **Note:** In the above solution we used the concept of divisibility rule which is a heuristic for determining whether a positive integer can be evenly divided by another (i.e. there is no remainder leftover). For example, determining if a number is even is as simple as checking to see if its last digit is 2, 4, 6, 8, or 0.