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M-Properties

1. Introduction

Integers can either be odd or even. Even numbers are multiples of the number two. Odd numbers are the rest.
Let even numbers have the property m2 (multiple of two).
Odd numbers will have the property m2 + 1.

What about numbers that are multiples of three or more?

Numbers that are multiples of three will have the property m3.
Numbers that are not multiples of three will have the property m3 + 1 or m3 + 2.

Numbers that are multiples of four will have the property m4.
Numbers that are not multiples of four will have the property m4 + 1, m4 + 3 or m4 + 3.
I call this class of properties M-Properties.
Define a set of numbers: mn, mn + 1, ... mn + (n - 1) as the mn set.

2. Adding numbers in the same mn set

The m2 set

The resulting properties are shown in the shaded part of the following table.
plus m2 m2 + 1
m2 + 1 m2 + 1 m2
m2 m2 m2 + 1

The m3 set

The resulting properties are shown in the shaded part of the following table.
plus m3 m3 + 1 m3 + 2
m3 + 2 m3 + 2 m3 m3 + 1
m3 + 1 m3 + 1 m3 + 2 m3
m3 m3 m3 + 1 m3 + 2

To find the property of the result of an addition

Leave the 'mn' part of the property unchanged and add the numeric parts.
  • If the resulting number is greater than n, subtract n from it repeatedly until the numeric part is less than n.
For example add an (m3 + 2) number to another (m3 + 2) number.
(m3 + 2) + (m3 + 2) = (m3 + 4) = (m3 + 1)

3. Multiplying numbers in the same mn set

The m2 set

The resulting properties are shown in the shaded part of the following table.
times m2 m2 + 1
m2 + 1 m2 m2 + 1
m2 m2 m2

The m3 set

The resulting properties are shown in the shaded part of the following table.
times m3 m3 + 1 m3 + 2
m3 + 2 m3 m3 + 2 m3 + 1
m3 + 1 m3 m3 + 1 m3 + 2
m3 m3 m3 m3

To find the property of the result of a multiplication

Leave the 'mn' part of the property unchanged and multiply the numeric parts.
  • If the resulting number is greater than n, subtract n from it repeatedly until the numeric part is less than n.
For example, multiply an (m3 + 2) number by another (m3 + 2) number.
(m3 + 2) × (m3 + 2) = (m3 + 4) = (m3 + 1)

4. Using mn algebraically

The m in the previous expressions is simply a descriptor meaning 'Any multiple of...'. Providing certain rules are obeyed it can also be used algebraically, to represent a number, as a multiplier.

Rules

  • mn + mn = mn (e.g. a multiple of 5 plus another multiple of 5 equals yet another multiple of 5)
  • mn - mn = mn
  • mn x mn = mn

Example

(m5 + 2)(m5 + 3) = (m5 x m5) + (3m5 + 2m5) + 6
= m5 + 5m5 + 6
= m5 + m5 + 6
= m5 + 6
= m5 + 1

  • Note: mn/mn does not always equal mn and should be avoided.
The tables on the previous pages were constructed by letting m = 1 and simply adding or multiplying the expressions.

5. Larger tables

The m8 set
times m8 m8 + 1 m8 + 2 m8 + 3 m8 + 4 m8 + 5 m8 + 6 m8 + 7
m8 + 7 m8 m8 + 7 m8 + 6 m8 + 5 m8 + 4 m8 + 3 m8 + 2 m8 + 1
m8 + 6 m8 m8 + 6 m8 + 4 m8 + 2 m8 m8 + 6 m8 + 4 m8 + 2
m8 + 5 m8 m8 + 5 m8 + 2 m8 + 7 m8 + 4 m8 + 1 m8 + 6 m8 + 3
m8 + 4 m8 m8 + 4 m8 m8 + 4 m8 m8 + 4 m8 m8 + 4
m8 + 3 m8 m8 + 3 m8 + 6 m8 + 1 m8 + 4 m8 + 7 m8 + 2 m8 + 5
m8 + 2 m8 m8 + 2 m8 + 4 m8 + 6 m8 m8 + 2 m8 + 4 m8 + 6
m8 + 1 m8 m8 + 1 m8 + 2 m8 + 3 m8 + 4 m8 + 5 m8 + 6 m8 + 7
m8 m8 m8 m8 m8 m8 m8 m8 m8
There are patterns in there but they are difficult to see. The patterns can be made clearer by the use of shading.

6. Shaded Tables

Replace the numbers in the table by shades of grey.















      m8       m8 + 1    m8 + 2    m8 + 3    m8+ 4    m8 + 5    m8 + 6    m8 + 7
times m8 m8 + 1 m8 + 2 m8 + 3 m8 + 4 m8 + 5 m8 + 6 m8 + 7
m8 + 7 m8






m8 + 6 m8


m8


m8 + 5 m8






m8 + 4 m8
m8
m8
m8
m8 + 3 m8






m8 + 2 m8


m8


m8 + 1 m8






m8 m8 m8 m8 m8 m8 m8 m8 m8

7. Shaded Patterns

Just show the shading.
 






 






 






 






 






 






 






 






The 'principal diagonal' runs from bottom left to top right.

Shaded Patterns

Multiplication tables - m8 and m9 sets
 

















      mn       mn + 1    mn + 2    mn + 3     mn + 4    mn + 5     mn + 6    mn + 7     mn + 8
 






 






 






 






 






 






 






 






 







 







 







 







 







 







 







 







 







m8
This is the pattern of a set where n is the multiple of two m2 numbers; in this case 2 x 4. The principal diagonal is divided into two equal parts by the black squares (mn).
m9 
This is the pattern of a set where n is a multiple of two m3 numbers; in this case 3 x 3.  The principal diagonal is divided into three equal parts by the black squares (mn).

Shaded Patterns

Multiplication tables - m10 and m11 sets
 





















      mn        mn + 1     mn + 2    mn + 3    mn + 4    mn + 5    mn + 6    mn + 7    mn + 8    mn + 9   mn + 10
 








 








 








 








 








 








 








 








 








 








 









 









 









 









 









 









 









 









 









 









 









m10
This is the pattern of a set where n is the multiple of two different primes; in this case 2 x 5. There is only one instance of a black square (mn). on the principal diagonal.

m11
This is the pattern of a prime number set. There are no instances of a black square (mn) except on the mn row and column.

Mike Holden - Sep 2005
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