Divisibility by Nine Number Trick

A well known divisibility trick exists to tell if a number is divisible by 9. First add all the digits of that number and if the result is divisible by 9, then that number is also divisible by 9. How does one prove this?

Elementary Number Theory (Springer Undergraduate Mathematics Series)

For convenience let us assume it is a 4 digit number we are testing, say \(abcd\). This can be expressed as
$$
\text{abcd} = 1d + 10c + 100b + 1000a
$$
This in turn can be expressed as
$$
d + c  + b + a + 9c + 99b + 999a \\
\text{or}\\
d + c + b + a + 9\times(c + 11b + 111a)\\
$$
Note that the second part of the number \(9(c + 11b + 111a)\) is divisible by 9. If \(d+c+b+a\) is divisible by 9, then the whole number is divisible by 9. Hence the proof. By extension, as 9 is divisible by 3, the method applies to 3 as well.
The following are some good books on learning probability

Book	Notes/Comments
Fifty Challenging Problems in Probability with Solutions (Dover Books on Mathematics)	This book is a great compilation that covers quite a bit of puzzles. What I like about these puzzles are that they are all tractable and don't require too much advanced mathematics to solve.
Introduction to Algorithms	This is a book on algorithms, some of them are probabilistic. But the book is a must have for students, job candidates even full time engineers & data scientists
Introduction to Probability Theory	Overall an excellent book to learn probability, well recommended for undergrads and graduate students
An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd Edition	This is a two volume book and the first volume is what will likely interest a beginner because it covers discrete probability. The book tends to treat probability as a theory on its own
The Probability Tutoring Book: An Intuitive Course for Engineers and Scientists (and Everyone Else!)	A good book for graduate level classes: has some practice problems in them which is a good thing. But that doesn't make this book any less of buy for the beginner.
Introduction to Probability, 2nd Edition	A good book to own. Does not require prior knowledge of other areas, but the book is a bit low on worked out examples.
Bundle of Algorithms in Java, Third Edition, Parts 1-5: Fundamentals, Data Structures, Sorting, Searching, and Graph Algorithms (3rd Edition) (Pts. 1-5)	An excellent resource (students, engineers and even entrepreneurs) if you are looking for some code that you can take and implement directly on the job
Understanding Probability: Chance Rules in Everyday Life	This is a great book to own. The second half of the book may require some knowledge of calculus. It appears to be the right mix for someone who wants to learn but doesn't want to be scared with the "lemmas"
Data Mining: Practical Machine Learning Tools and Techniques, Third Edition (The Morgan Kaufmann Series in Data Management Systems)	This one is a must have if you want to learn machine learning. The book is beautifully written and ideal for the engineer/student who doesn't want to get too much into the details of a machine learned approach but wants a working knowledge of it. There are some great examples and test data in the text book too.
Discovering Statistics Using R	This is a good book if you are new to statistics & probability while simultaneously getting started with a programming language. The book supports R and is written in a casual humorous way making it an easy read. Great for beginners. Some of the data on the companion website could be missing.
A Course in Probability Theory, Third Edition	Covered in this book are the central limit theorem and other graduate topics in probability. You will need to brush up on some mathematics before you dive in but most of that can be done online
Probability and Statistics (4th Edition)	This book has been yellow-flagged with some issues: including sequencing of content that could be an issue. But otherwise its good