Test for Divisibility of Numbers

What is divisibility?

In the following number, replace $*$ by smallest digit and largest digit to make the number divisible by $3$.

$*4129$

In the following number, replace $*$ by smallest digit and largest digit to make the number divisible by $3$.

$205*3$

In the following number, replace $*$ by smallest digit and largest digit to make the number divisible by $3$.

$3*215$

If the eight-digit number $2575d568$ is divisible by $54$ and $87$, the value of the digit '$d$' is

Find the maximum value of digit $\mathrm{z}$, if $47\mathrm{z}592$ is divisible by $3$.

If $928\mathrm{z}89$ is divisible by $11$, where $\mathrm{z}$ is a digit, what is value of $\mathrm{z}$?

If $869\mathrm{z}84$ is divisible by $11$, where $\mathrm{z}$ is a digit, what is value of $\mathrm{z}$?

Prove the following property of divisibility:

If $a,b,$ and $c$ are three natural numbers such that $a$ is divisible by $b$, and $b$ is divisible by $c$, then $a$ is divisible by $c$ also.

Explain the following property with the help of an example

If $a,b,$ and $c$ are three natural numbers such that $a$ is divisible by $b$, and $b$ is divisible by $c$, then $a$ is divisible by $c$ also.

Explain the following property with the help of an example

If a number is divisible by another number, then it is divisible by each of the factors of that number.

Create two different four-digit numbers that are divisible by: 

$4, 6$ and $9$

The number $10080$ is completely divisible by $2,3,4,5,6,7,8,9$ respectively.