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{aligned}
(1+x)^{\mathrm{n}}-1 & =\left({ }^{\mathrm{n}} \mathrm{C}_0+{ }^{\mathrm{n}} \mathrm{C}_1 \mathrm{x}+{ }^{\mathrm{n}} \mathrm{C}_2 \mathrm{x}^2+\ldots \ldots+{ }^{\mathrm{n}} \mathrm{C}_{\mathrm{n}} \mathrm{x}^{\mathrm{n}}\right)-1 \\
& =\mathrm{x}\left[{ }^{\mathrm{n}} \mathrm{C}_1+{ }^{\mathrm{n}} \mathrm{C}_2 \mathrm{x}+\ldots \ldots .+{ }^{\mathrm{n}} \mathrm{C}_{\mathrm{n}} \mathrm{x}^{\mathrm{n}-1}\right]
\end{aligned}
$

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

$
\begin{aligned}
(1+x)^n-n x-1 & =\left({ }^n C_0+{ }^n C_1 x+{ }^n C_2 x^2+\ldots \ldots+{ }^n C_n x^n\right)-n x-1 \\
& =x^2\left[{ }^n C_2+{ }^n C_3 x+\ldots \ldots+{ }^n C_n x^{n-2}\right]
\end{aligned}
$
For example
Prove that $3^{2 n+2}-8 n-9$ is divisible by 8 if $n \in N$

$
\begin{aligned}
3^{2 n+2} & -8 n-9=(1+8)^{n+1}-8 n-9 \\
& =\left[1+(n+1) 8+{ }^{n+1} C_2 8^2+\ldots\right]-8 n-9 \\
& ={ }^{n+1} C_2 8^2+{ }^{n+1} C_3 8^3+{ }^{n+1} C_4 8^4+\ldots \\
& =8\left[{ }^{n+1} C_2 8+{ }^{n+1} C_3 8^2+{ }^{n+1} C_4 8^3+\ldots\right]
\end{aligned}
$

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 $(15-7)=8$.