**Introduction**

The world around us is amazing with a variety of different mysteries. One of the keys to the mysteries of the universe is numbers. As far back as Pythagoras said: “The world is built on the power of numbers.”

In my school course I studied the signs of divisibility by 2, 3, 5, 9, 10. This is not enough and I decided to expand my knowledge on this subject. That is how the work on this topic began.

**Relevance**

In studying the topic: “The signs of the divisibility of natural numbers by 2, 3, 5, 9, 10” we were interested in the question of the divisibility of numbers. It is known that not always one natural number is divisible by another natural number without a remainder. When dividing natural numbers, we get a remainder, we make mistakes, as a result – we lose time. The signs of divisibility help, without doing division, to determine whether one natural number is divisible by another natural number. We decided to write a research paper on this topic.

To supplement the already known signs of the divisibility of natural numbers studied at school and to add to their knowledge of the signs of the divisibility of numbers.

** Objectives **

1. To investigate additional literature that supports the hypothesis of the existence of other characteristics of divisibility of positive integers and the validity of our identified characteristics of divisibility.

2. Write out the signs of divisibility of natural numbers in 7, 11, 12, 13, 14, 19, 37 found from additional literature.

3. Draw a conclusion.

**Hypothesis **

If the divisibility of natural numbers by 2, 3, 5, 9, 10 can be determined, then there must be signs by which the divisibility of natural numbers by other numbers can be determined.

**Theoretical material**

We encounter numbers at every step, they accompany us from birth to old age.

At school in math classes, we study different numbers and perform operations with them. Often we have to do division to solve a problem.

From mathematics lessons, I learned about the divisibility of numbers by 2, 3, 5, 9 and 10. According to these rules I can now, without doing division, find out whether a given number is a multiple of, for example, two or nine.

In the decomposition of numbers into prime factors I had to divide the number by 7. How do I know if I am dividing with or without a remainder? Are there no signs of divisibility by 7? What about other numbers?

I found a definition of the divisibility sign in an encyclopedic dictionary.

Divisibility is a rule that allows you to quickly determine whether a number is a multiple of one or another number.

With the help of the Internet and additional literature, I have found the signs of divisibility of numbers, which I have not studied in school program:

1) A number is divisible by 7 when the result of subtracting the doubled last digit from that number without the last digit is divisible by 7

(For example, 343 is divisible by 7 because 34- (2*3) =34 – 6=28 is divisible by 7).

I also discovered another way:

2) To find out whether the number is divisible by 7 , divide it from the right hand into faces of 3 digits in each; make the sum of the faces of odd order and from them subtract the sum of the faces of even order (or vice versa). If the difference is divisible by 7, then the number is divisible by 7.

While working on this topic, I learned that there are a couple of very different signs of divisibility by 11.

1) If the sum of the digits in the even places is equal to or greater than the sum of the digits in the odd places, then the number is divisible by 11.

(For example, 14641 is divisible by 11 because 1+6+1=4+4.)

2) The number is divided from right to left into groups of two digits in each group and the groups are added. If the sum obtained is a multiple of 11, then the number is a multiple of 11.

In order to find out whether the number is a multiple of 12, you should check whether it is divisible by 4 and by 3 at the same time.

(For example, 24 is divisible by 4 and by 3, so it would be a multiple of 12.)

So far I have only found one sign of divisibility by 13. It is similar to divisibility by 7, only the last digit should not be doubled, but multiplied by 9.

(For example, 361 is divisible by 9, because 36 – (1*9) = 36 – 9 = 27 is divisible by 9.)

The number that is divisible by 14 can be tested by the same scheme as in 12. Only the number must be divisible by 7 and by 2.

(For example, the number 42 is simultaneously divisible by 2 and 7).

A number is divisible by 19 if and only if the number of its tens added to the doubled number of ones is a multiple of 19.

(For example, 646 is divisible by 19 because 64 + (6 × 2) = 76 is divisible by 19.)

A number is divisible by 37 if and only if, when you divide a number into groups of three digits (starting with ones), the sum of those groups is a multiple of 37.

**Research**

Signs of divisibility apply to a variety of number focuses. For example:

One person writes down any three-digit number on a piece of paper. Passes it to the other person. The second person adds the same number to the right side of the number and passes this entry of a six-digit number to the third person. Let the third divide this number by 7 and pass it to the fourth. The fourth will divide the result by 11 and pass it on to the fifth. The fifth will divide the result by 13 and pass it to the first. If all the calculations have been done correctly, the first will get the three-digit number he originally wrote on the paper. The surprise in this trick is not that the first one gets the number he wrote down, but that the “trickster” is sure that the number in question is divisible by 7, 11, 13 – which is not very common. The clue is that by adding exactly the same three-digit number to a three-digit number, it is equal to multiplying by 1001. And 1001 equals the product of 7, 11, 13.

I decided to do a survey of my classmates to see if they know the signs of divisibility, such as 7, 11, 33, 14, and whether they want to learn them. The boys’ answers pleased me. It turned out that most of them, like me, want to learn more about the signs of divisibility, this topic aroused their interest.

After learning many other signs of divisibility, I made a conclusion:

Working with different sources, I became convinced that there are other signs of divisibility of natural numbers (by 7, 11, 12, 13, 14, 19, 37), which confirmed the correctness of the hypothesis that there are other signs of divisibility of natural numbers.

Knowing and using the above signs of the divisibility of natural numbers greatly simplifies many calculations, thereby saving time; eliminating computational errors that can be made when performing the action of division. It should be noted that the wording of some of the signs is difficult. Maybe that’s why they are not taught at school.

**Conclusion**

After getting acquainted with the signs of divisibility of numbers, I believe that I can use the acquired knowledge in my own learning activities, independently apply this or that sign to a particular problem, apply the studied signs in a real situation.

I searched for information on this topic in an encyclopedia and online sources. Everything I found will help me in solving the examples, make my work easier and faster.