# Proving divisibility rules?

I've realized that a lot of textbook questions require me to google things because I have no clue how to prove certain things.

For example, I do not have the fact that if the last 2 digits in a number are divisible by 4, that number is then divisible by 4.

I'm pretty sure my teacher will not expect me to have this memorized, and my course definitively doesn't need me to be able to prove it... but I want to be able to.

At what point will be able to do this.. currently I am reading basic mathematics by lang. How many more books would I have to read? Can anyone recommend me a book after langs basic math book? I was thinking of Spivak calculus but idk man. I just find it depressing that I'm not able to prove this right now.

andrewkirk
Homework Helper
Gold Member
Each of the divisibility rules is fairly easy to prove. Most use mathematical induction (eg the rules for 3 and 9). But the one you mention is even easier.

Let n be the number and let the digits excluding the last two make number p and the last two make number q.
Then we have

n = 100p + q

Now 100 is divisible by 4, so 100p must be as well. So n is divisible by 4 if and only if q - the number made from the last two digits - is divisible by 4.

• Each of the divisibility rules is fairly easy to prove. Most use mathematical induction (eg the rules for 3 and 9). But the one you mention is even easier.

Let n be the number and let the digits excluding the last two make number p and the last two make number q.
Then we have

n = 100p + q

Now 100 is divisible by 4, so 100p must be as well. So n is divisible by 4 if and only if q - the number made from the last two digits - is divisible by 4.

That really does seem like a simple proof. Do you know what book I should read after basic mathematics? I want a solid foundation in math so that I can be kind of like a "jack of all trades" and learn topics from other fields like computer science, physics, engineering. I'm just now getting through basic mathematics but I hope to be done with the book in about a month...

fresh_42
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2021 Award
That really does seem like a simple proof. Do you know what book I should read after basic mathematics? I want a solid foundation in math so that I can be kind of like a "jack of all trades" and learn topics from other fields like computer science, physics, engineering. I'm just now getting through basic mathematics but I hope to be done with the book in about a month...
If you want to read those things in a book, then the first row here: https://openstax.org/subjects is a good source. Such things should be included in regular school books, which you normally don't read cover to cover, so reading all of them might be over the top. But at least these books are free, recommendable and you can look beforehand what you want to practice, resp. which chapters are relevant to you.

andrewkirk