Divisibility Rules

Question

What are some ways that your teacher has shown divisibility rules? How do you know to divide by a certain number? So many students have learned different ways and I would love to hear yours!

Answer 1

The way that I teach what numbers a certain number is divisible by is through an example that I got from a magazine article.

Is the number 3599 a prime number? I like this example.Because you are not only trying to decide if a certain number.Divides evenly.Into 3599.Comma.You are having to figure out.Any numbers that might divide into 3599?Here is how I go about it.Colon.

1. Is 3599 an even number? No, and therefore.It is not divisible by any even number.
2. The sum of the digits of the number 3599 is 26. Is 26 divisible by three? The answer is no. If it was, then 3599 would also be divisible by three. Also, if 26 was divisible by 9, then 3599 would also be divisible by 9.
3. Does 3599 end in 5? No, but if it did then 3599 would be divisible by 5.
4. Does 3599 end in zero? No, but if it did then 3599 would be divisible by 10.
5. The first four steps are the basic way that I start to decide what factors there are of a given number. Because I found no factors aside from 1 and 3599, I was close to calling this a prime number.  
6. Before I did, I noticed that this particular number is very close to the perfect square 3600.In fact.You could write the equation:

3599=3600-1

From here, notice you can rewrite this as

3599=60^2-1^2

This gives us a difference of squares. Remember that rule says:

a^2-b^2 = (a+b)(a-b), so

3599=60^2-1^2=(60+1)(60-1)=61*59

Therefore 3599 is not a prime number, because it has factors other than one and itself.

There is no perfect way to find what factors a number has, but the above steps should be helpful, especially numbers one through four.

Answer 2

Answer 3

In order to know that a number is prime, you need to try all the prime divisors up to the square root of the number in which you are interested.

For relatively small numbers this is not too onerous. As the numbers get bigger, it becomes at the very least a nuisance!

For really big numbers there are sieves, the simplest (and perhaps best known) is the sieve of Eratosthenes for which there is a good description in Wikipedia.

Answer 4

change division to multiplication

invert the divisor

multiplying is easier than dividing