It is known that when we are factoring a certain number and we verify that it is not divisible by 2, the next prime number that we are going to test is 3, so we must also know the divisibility criteria of this number.
The criterion for divisibility by 3, unlike the criterion for number 2, depends on a relationship between all the digits of the number to be divided. Let's see what this relationship should be:
"In order for a number to be divisible by the prime number 3, the sum of the digits of this number must be divisible by 3."
For a better understanding, let's look at an example: let's see if the number 234 is divisible by 3.
The sum of the digits that make up the number 234 é: 2+3+4 = 9. It's much easier to know if the number 9 can be divided by 3 than the number 234. Like nine (number that resulted from the sum of the digits of the number 234) can be divided by 3, we can say that the number 234 is divisible by 3.
Therefore, to check the divisibility by 3 we must pay attention to all the digits, add them carefully and check if the sum is, in fact, divisible by 3. Note that in this criterion you, after adding the figures, must perform a division by the number 3, however, it is a much simpler division, let's see a proof of this fact.
Verify the number 134193621 is divisible by 3.
If we were to divide this number by three, we would definitely spend good lines of calculation, but we saw previously, it is enough to add the digits of this number to obtain the divisibility answer by 3.
Adding the digits: 1+3+4+1+9+3+6+2+1 = 30.
If the sum of these digits is divisible by 3, we can say that the number 134193621 is actually divisible by 3. It's very easy to check the divisibility of the number 30 by 3, isn't it? 30 divided by 3 equals 10, an exact division.
Remember that the process we've done is only to check whether the division of the number 134193621 is divisible by 3, this does not mean that the value 10 is the result of dividing this number by three.
By Gabriel Alessandro de Oliveira
Graduated in Mathematics
Brazil School Team