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
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.
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.
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.
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.
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.
Signs of