Exploring Divisibility Rules in Mathematics for Self-Study

In summary, the conversation is about the difficulty of proving certain mathematical concepts and the desire to have a solid foundation in math in order to pursue other fields such as computer science, physics, and engineering. The conversation also includes a simple proof for the divisibility rule for 4 and a recommendation for the book "Spivak calculus" as a good source for building a strong mathematical foundation.
  • #1
Rijad Hadzic
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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.
 
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  • #2
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.
 
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  • #3
andrewkirk said:
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...
 
  • #4
Rijad Hadzic said:
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.
 
  • #5
Rijad Hadzic said:
I was thinking of Spivak calculus
That text is my all-time favourite mathematics book. I would strongly recommend it. It's easy to follow and requires very little prior knowledge.
 

FAQ: Exploring Divisibility Rules in Mathematics for Self-Study

1. How do you prove the divisibility rule for 2?

The divisibility rule for 2 states that a number is divisible by 2 if the last digit is even. To prove this, we can use the concept of even and odd numbers. Since every even number can be divided by 2 without a remainder, we can conclude that any number with an even last digit is also divisible by 2.

2. What is the proof for the divisibility rule for 3?

The divisibility rule for 3 states that a number is divisible by 3 if the sum of its digits is also divisible by 3. To prove this, we can use the concept of modular arithmetic. We can represent a number as the sum of its digits multiplied by their place values. Since any multiple of 3 can be divided by 3 without a remainder, we can conclude that the sum of the digits must also be divisible by 3.

3. Can you explain the proof for the divisibility rule for 9?

The divisibility rule for 9 states that a number is divisible by 9 if the sum of its digits is also divisible by 9. This rule is similar to the rule for 3, but instead of considering multiples of 3, we consider multiples of 9. We can also use modular arithmetic to prove this rule, by representing a number as the sum of its digits multiplied by their place values. Since any multiple of 9 can be divided by 9 without a remainder, we can conclude that the sum of the digits must also be divisible by 9.

4. How can you prove the divisibility rule for 5?

The divisibility rule for 5 states that a number is divisible by 5 if the last digit is either 0 or 5. To prove this, we can again use the concept of modular arithmetic. Since every multiple of 5 has a last digit of either 0 or 5, we can conclude that any number with a last digit of 0 or 5 is also divisible by 5.

5. What is the proof for the divisibility rule for 11?

The divisibility rule for 11 states that a number is divisible by 11 if the difference between the sum of its digits in even place values and odd place values is either 0 or a multiple of 11. To prove this, we can once again use modular arithmetic. We can represent a number as the sum of its digits multiplied by their place values, and then rearrange the terms to show that the difference between the even and odd place values is divisible by 11.

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