Divider image
The divider picture (also divider diagram ) is a diagram in which all the divisors of a number are shown according to a certain scheme. The prime factors play a special role in this scheme . The divider is a special variant of the Hasse diagram .
Structure of a divider image
see picture with example 18 .
- In the bottom row always is 1 .
- In the 2nd row are the prime factors of the number.
- In the example the 2 and 3.
- In the 3rd row the divisors of the number, which are composed of 2 prime factors .
- in the example the 6 (= 2 · 3) and the 9 (= 3 · 3).
- This is followed by the series with 3, 4 etc. prime factors,
- in the top row is the number that is involved (here 18 ).
Each divider in the picture is related to its divisors and multiples in the adjacent rows. Symbolize in the picture
- the blue lines a multiplication by 2 ,
- the red lines a multiplication by 3 .
So on the way from 1 to 18, two red lines and one blue line are always used, which corresponds to the calculation 2 · 3 · 3 = 18, or 3 · 3 · 2 or 3 · 2 · 3.
Create the divider image using the prime factors
(see example image: number 18)
- By prime factorization determining the prime factors of the number sought.
- In the bottom row you write the 1 , in the 2nd row the prime factors , in the example the 2 and the 3 .
- The prime factors are connected to the 1 by lines . The line from 1 to 2 (blue line in the example) now means "take 2" , the line from 1 to 3 (red) means "take 3" etc.
- From the 2nd row you get to the 3rd row by adding more lines to the dividers . So the 3 with a blue line results in the 6 (“take 2”) and with a red line the 9 (“take 3”). From the 2 there is also the 6 with a red line. No further blue line may be added to the 2, since the 2 only occurs once in the prime factorization of 18.
Now you get from the 1 via 2 red lines (3 · 3) to the 9 , as well as via red, blue (3 · 2) to the 6 and via blue, red (2 · 3) also to the 6.
- The 4th row continues accordingly. Only the blue line can connect to the 9, since both 3s on the way from 1 to 9 have already been used up and only the 2 is left. This leads to the 18 (9 · 2). Only the red line may follow at the 6, as the only available 2 has already been used up on the way to the 6. Also 6 3 equals 18 .
Creating the divider image - playful method
- By prime factorization determining the prime factors of the number sought.
- You write each prime factor on a card and put all cards into a lottery drum (cup or similar).
- At 18 you get 3 cards, one labeled 2 and two labeled 3 .
- The divider picture starts with 1 in the bottom row .
- The 2nd row: You write down all the numbers that are actually shown on the cards.
- For 18, these are 2 and 3 . The 3 is only written once in the divider image, even if it occurs twice in the 18 (2 * 3 * 3).
- The 3rd row: One blind draws two cards , the numbers taking each time and writes the result to the third row. Repeat this a few times until you have all the dividers in the 3rd row together.
- So you get the 9 (3 · 3) and the 6 (3 · 2). You don't get the 4 (2 · 2) because the 2 is only once in the pot. It is also not a factor of 18.
- The 4th row: Here takes you 3 cards and multiply them, and writes the results.
- In the example you get the 18 . These are all cards in the pot and the game is over.
- then you connect all numbers that are divisors or multiples of each other if they are in adjacent rows.
additional
Structured divider image and calculation template
When creating the splitter image, it is a good idea to have all lines that indicate a particular operation (for example, all "x2" lines) in the same direction and length. This creates a structured picture. In addition, you can now create a calculation template, as indicated in the example images. If you put the "X" on a number in the divider image, you get the right result at the end of the lines.
Part-divider images
If you have already made a picture of a large number and then need the picture of a number that is a divisor of the first number, you can use the divisor diagram of the large number by deleting some numbers.
Example: You already have the divider image of 360. If you now need the image of 180, you remove the numbers 360, 120, 40, 72, 24 and 8 from the 360 image and it's done.
Extend divider images
Conversely, a picture of a small number can be made into a large number by "adding something to it". To do this, the large number must be a multiple of the small number.
Example: For a picture of the 720, take the 360 picture again and extend it beyond the 8 with a blue line: 8 · 2 = 16. The same thing happens at 360, 120, 40, 72 and 24.
3D divider images
Divider images with more than 2 different prime factors quickly become confusing because many lines cross. This can be avoided by building a three-dimensional model (e.g. from spheres and rods). With a little imagination, the cuboid structure of the 360 image can be seen in the example.