Chinese Remainder Theorem

Create a numerical example of the Chinese Remainder Theorem with numbers different from those in the Youtube below.  

Introduction to the Chinese Remainder Theorem

Some time in the first century AD a Chinese mathematician by the name of Sun Zi published a book, Sun Zi Suanjing, or "The Arithmetical Classic of Sun Zi". In this book Sun Zi introduced a method of solving systems of linear congruences that became known as the Chinese Remainder Theorem.

The Chinese Remainder Theorem (abbreviated as "CRT") is easily understood with a riddle: 

An old woman goes to the market with a basket of eggs. She sets the basket down and a horse accidentally steps on it, crushing all the eggs. The rider offers to pay her for the damaged eggs and asks how many eggs did she have. She tells the rider that she cannot remember but that when she had taken all of the eggs out 7 at a time, there was 1 left in the basket. When she took them out 4 at a time, there were 3 left and when she took them out 5 at a time, there were 2 left. What is the smallest number of eggs she could have had?

Example of the Chinese Remainder Theoremhttps://www.youtube.com/watch?v=ru7mWZJlRQg