Problems on Divisibility - Practice Questions & MCQ

Given Expression is divisible by an Integer
1. Expression, $(1+\mathrm{x}{)}^{\mathrm{n}}-1$ is divisible by x because

$\begin{array}{rl}(1+x{)}^{\mathrm{n}}-1& =({}^{\mathrm{n}}{\mathrm{C}}_0+{}^{\mathrm{n}}{\mathrm{C}}_1\mathrm{x}+{}^{\mathrm{n}}{\mathrm{C}}_2{\mathrm{x}}^2+\dots \dots +{}^{\mathrm{n}}{\mathrm{C}}_{\mathrm{n}}{\mathrm{x}}^{\mathrm{n}})-1\\ & =\mathrm{x}[{}^{\mathrm{n}}{\mathrm{C}}_1+{}^{\mathrm{n}}{\mathrm{C}}_2\mathrm{x}+\dots \dots .+{}^{\mathrm{n}}{\mathrm{C}}_{\mathrm{n}}{\mathrm{x}}^{\mathrm{n}-1}]\end{array}$

2. Expression, $(1+\mathrm{x}{)}^{\mathrm{n}}-\mathrm{nx}-1$ is divisible by ${\mathrm{x}}^2$ because

$\begin{array}{rl}(1+x{)}^n-nx-1& =({}^n{C}_0+{}^n{C}_{1}x+{}^n{C}_2{x}^2+\dots \dots +{}^n{C}_n{x}^n)-nx-1\\ & =x^2[{}^n{C}_2+{}^n{C}_{3}x+\dots \dots +{}^n{C}_n{x}^{n-2}]\end{array}$
For example
Prove that $3^{2n+2}-8n-9$ is divisible by 8 if $n\in N$

$\begin{array}{rl}{3}^{2n+2}& -8n-9=(1+8{)}^{n+1}-8n-9\\ & =[1+(n+1)8+{}^{n+1}{C}_2{8}^2+\dots ]-8n-9\\ & ={}^{n+1}{C}_2{8}^2+{}^{n+1}{C}_3{8}^3+{}^{n+1}{C}_4{8}^4+\dots \\ & =8[{}^{n+1}{C}_{2}8+{}^{n+1}{C}_3{8}^2+{}^{n+1}{C}_4{8}^3+\dots ]\end{array}$

Which is divisible by 8.

Divisibility: Important Results

1. The expression ${\mathrm{a}}^{\mathrm{n}}-{\mathrm{b}}^{\mathrm{n}}$ is divisible by $\mathrm{a}+\mathrm{b}$, if n is even.
2. The expression ${\mathrm{a}}^{\mathrm{n}}-{\mathrm{b}}^{\mathrm{n}}$ is divisible by $\mathrm{a}-\mathrm{b}$, if n is even or odd.
3. The expression ${\mathrm{a}}^{\mathrm{n}}+{\mathrm{b}}^{\mathrm{n}}$ is divisible by , $\mathrm{a}+\mathrm{b}$ if n is odd.

In all the above cases n is a natural number.
For Example
The expression ${15}^4-7^4$ is divisible by $(15+7)=22$ and $(15-7)=8$.