NSW Y12 Maths - Extension 1 Proof Divisibility

Divisibility Videos How to buy Buy Books

Resources for Divisibility

Questions

15
With Worked Solution

Click Here
Video Tutorials

1

Click Here
HSC Questions

2
With Worked Solution

Click Here

Divisibility Theory

$The statement $S(n)=m J$ where $J$ is an integer, implies that $S(n)$ is divisible by $m$.\\ The standard rules of mathematical induction are applied as in the example.\\ \begin{multicols}{2} \textbf{Example}\\ %118310 Prove that $3^{3n}+2^{n+2}$ is divisible by $5$ for all positive integers $n$. \\ \textbf{Solution}\\ $\text { To prove } 3^{3 n}+2^{n+2}=5 J \quad(I)$\\ $\begin{aligned} \text{For } n=1 \quad\text { LHS } &=3^3+2^3 \\ &=27+8 \\ &=35 \\ &=5 \mathrm{~J} \quad(J=7)\\ \therefore &(I) \text{ is true for }n=1 \end{aligned}$\\ $\text { Assume (I) is true for } n=k$\\ $\text { i.e. } \left.\quad 3^{3 k}+2^{k+2}=5 J^{\prime} \text { ( } J^{\prime} \text { an integer }\right)\quad \text { (II) }$\\ $\text { Prove (I) is true for } n=k+1$\\ $\text { i.e } 3^{3 k+3}+2^{k+3}=5 J^{\prime \prime}\left(J^{\prime \prime}\right. \text { an integer) } \quad \text{(III)}$\\ \columnbreak $\begin{aligned} \text{In III} \quad L H S&=3^{3 k+3}+2^{k+3}\\ \text{In (II)} \quad 3^{3k}&=5 J^{\prime}-2^2 \times 2^k\\ \text { LHS }&=3^3 \times 3^{3 k}+2^3 \times 2^k\\ &=27\left(5 J^{\prime}-4 \times 2^k\right)+8 \times 2^k\\ &=27 \times 5 J^{\prime}-108 \times 2^k+8 \times 2^k\\ &=27 \times 5J^{\prime}-100 \times 2^k\\ &=5\left(27 J^{\prime}-20 \times 2^k\right)\\ &=5 J^{\prime \prime} \left(J^{\prime \prime}=27 J^{\prime}-20 \times 2^k \text { an integer }\right)\\ &=R H S\\ \therefore LHS&=\text { RHS }\\ &\therefore \text { frue for } n=k+1 \end{aligned}$\\ Now $(I)$ is true for $n=1$ so by mathematical induction (I) is true for all $n \geqslant 1$\\ \end{multicols}$

Create account

I am..

A teacher
A student
A tutor

Please enter your details

First Name.

Last Name

Email Address

State

I agree with your terms of service

Back to Topic (Proof)

Videos

Videos relating to Divisibility.

Divisibility - Video - Proof by Induction Divisibility proofs

Plans & Pricing

With all subscriptions, you will receive the below benefits and unlock all answers and fully worked solutions.

Teachers Tutors

Features

Free

Pro

All Content

All courses, all topics

Questions

Answers

Worked Solutions

System

Your own personal portal

Quizbuilder

Class Results

Student Results

Exam Revision

Revision by Topic

Practise Exams

Answers

Worked Solutions
Awesome Students

Features

Free

Pro

Content

Any course, any topic

Questions

Answers

Worked Solutions

Extended Response

Questions and Answers

Fully Worked Solutions

Videos

Curated Topic Videos

Premium Topic Videos

System

Your own personal portal

Basic Results

Analytics

Study Recommendations

Exam Revision

Revision by Topic

Practise Exams

Answers

Worked Solutions