Prime Factors: Part 2

Prime Factors:

The 3 family

We have taken it as a maxim that the sum of the digits will let us know if a number is divisible by 3.  But when we put it with its family, we see a clear pattern, while at the same time "proving" the maxim.

03 => 0+(3*1) = 3.

That is quite obvious, but using the formula on bigger numbers, removing the digit to the far right and adding it to  a tenth of the digits to its left, we can see WHY our maxim is true:

2853 => 285-9 = 276 => 27-18 = 9 (3*3)

2+8+5+3 = 18 (3*6); if in doubt, keep going: 18 => 1+8 = 9

Amazingly, the multipliers progress by 3, and in the third and fourth columns below, the family relationship is abundantly clear!


03 => 0+(3*1)   = 0+03 = 03
13 => 1+(3*4)   = 1+12 = 13
23 => 2+(3*7)   = 2+21 = 23
33 => 3+(3*10) = 3+30 = 33*
43 => 4+(3*13) = 4+39 = 43
53 => 5+(3*16) = 5+48 = 53
63 => 6+(3*19) = 6+57 = 63*
73 => 7+(3+22) = 7+66 = 73
83 => 8+(3*25) = 8+75 = 83
93 => 9+(3*28) = 9+84 = 93*

Building this table is easy, especially when you remember to add 3 to the multiplier, but then again, this is the "3 family."

Column 1, start with 0, add one per row.
Column 2, 3 all the way down
Column 3, just like #1 followed by "+"
Column 4, just like 2, but proceed with "("
Column 5, start with 1, add 3 per row
Column 6, same as #3
Column 7, start with 03, add 9 per row. Close with ")"
Column 7a repeats the 3 and goes on to 8
Column 7b starts with {1}3 and goes down to {0}4
Column 8, start with 03 and add 10 per row.

Wow.  I'm not sure if that is "easy" or not.  The second half presents us with a problem of large numbers as factors.  However, notice that one of them is 13 (in the family).

A second, easier, factor presents itself on down the line.  Since 30 is obviously in the family, and when added to 13 gives us 43, we have an alternative to multiplying by 13.

43 => 4-(3*30) = 4-90 = -86  (-2*43).

The other factors for primes -- 53, 73 and 83 -- use factorable numbers: 16, 22, and 25 (two squares and 2*11).  The 3 family is a math friendly family all around.

So, what do I know?

I know that fabulous patterns emerge if one looks closely.  These patterns are demonstratable using the properties of math, and can be proven.  However, it is much easier just to experiment with the results.


Here is the "chart" with only the primes:

03 => 0+(3*1)  = 0+03 = 03 (special case 1*)
13 => 1+(3*4)  = 1+12 = 13
23 => 2+(3*7)  = 2+21 = 23

43 => 4+(3*13) = 4+39 = 43 (special case 2*)
53 => 5+(3*16) = 5+48 = 53 (16 = 4*4)

73 => 7+(3+22) = 7+66 = 73 (22 = 2*11)
83 => 8+(3*25) = 8+75 = 83 (25 = 5*5)

*1 Adding the digits and reducing to a recognizable multiple of 3 works for this.  The square of 3 is 9, leaving a bonus indicator for divisibility by 9.  If the number is even, then both 2 and 3 are prime factors.

*2 Beyond 39 (3*13), so an alternative using 30 (3*10) presents itself:

43 => 4-(3*30) = 4-90 = -86 (-2*43)

Happy calculating!