Prime Factors: Part 5

Prime Factors:

The 19 Family

Rightfully, this should be called the "9" Family, but alas, 9 is not a prime number.  However, as a member of the 19 family, 9 come holds an honored place in mathematics.  In the  similar "warped mirror" 11 family the 1 performed a similar place of honor.  1 is not a prime, for it does not have another factor.

Like 1, whose difference from 0 is a +1, 9 is different from 10 by 1, but it is a negative difference.  As such, the key to discerning the 19 family of primes is seeing a difference of 1 per 10 following a plus sign.  As you may recall, with the 11 family the multiplier followed a minus sign.

And so, starting with 19, let's build that table:

19 => 1+(9*2) = 1+18 = 19
29 => 2+(9*3) = 2+27 = 29

and so on. I will throw in the "09" to show the full sequence.

09 => 0+(9*1) = 0+9  =  9 -- not prime
19 => 1+(9*2) = 1+18 = 19
29 => 2+(9*3) = 2+27 = 29
39 => 3+(9*4) = 3+36 = 39 -- not prime
49 => 4+(9*5) = 4+45 = 45 -- not prime
59 => 5+(9*6) = 5+54 = 59
69 => 6+(9*7) = 6+63 = 69 -- not prime
79 => 7+(9*8) = 7+72 = 79
89 => 8+(9*9) = 9+81 = 89
99 => 9+(9*10)= 9+90 = 99 -- not prime

Half of the family are not prime numbers.  This no surprise when the "father" of the family is the one odd number among single digits that is a square.  Note that there is one more square in the family: the number 49.  A blessed family indeed.

As before, this blog would not be complete if we did not have and example or two.  I will start with a random 5 digit number and a random family member.

The random number: 54,977 -- my finger bounced on the 7, but let's go with it. Seeing the 54, let's use 59.

54,977 = 5497+(7*6) = 5497+42=5539 => 553+(9*3)=553+27= 580

58 is not divisible by 59, but came so very close!

Now, I will take the number 5497 from step two above and multiply it time 59.

324,323 => 32432+(3*6)=32432+18=32450 => 32450=>3245
3245 => 324+(5*6) = 324+30 = 354 => 35+(4*6)= 35+24 = 59.

That took some time, but we didn't have to use a calculator.  Well, I used one to get the original number, but I didn't have to.  Working with the "9 family" all I had to do was multiply by 60 and subtract the original number.

      5,497

         x 60

   329,820

    - 5,497

   324,323

I love "short cuts."  Looking at the number another way, I could have taken 5500*59 - 3*59.  That is, (5500-3)(59).  But that would have taken longer.

Now, no more nonsense.  One more example, using a manufactured number multiplying some simple primes, the last one being an unknown member of the 19 family.


33,495/5 =
6,699/11 = 
609/3    =
203 => 20+(3*3) = 29.  203/29 = 7

5*11*3*7*29 = 33,495.

Okay, I could have used the test from the 7 family first, but  please note:

20+(3*x) = 29
           3x = 29-20 = 9
             x = 3

So, what do I know?

The patterns for the "families" of prime numbers are easy to discern.  There are four such families.  The 3 and 7 families are "mirror images" of each other, as are the 11 and 19 families.  I will include a neat chart on my next blog post.

For now, here is the chart for the "19" family.


19 => 1+(9*2) = 1+18 = 19
29 => 2+(9*3) = 2+27 = 29
59 => 5+(9*6) = 5+54 = 59
79 => 7+(9*8) = 7+72 = 79
89 => 8+(9*9) = 9+81 = 89

Note the form:

X+9 = (X/10)+[9*(X+1)]