Prime Factors: Part 3

Prime Numbers:

The Amazing Mr. Five.

Of the prime numbers under 10, only 5 stands alone.  He has no other primes in his family.  As such, recognizing multiples are easy. Just look for a 5 at the end.

This is true, because any number that is divisible by 10 -- that is has a 0 at the end -- divisible by both 2 and 5.  Further determination is made after removing the 0.

For instance:

1234560 is divisible by 2 and 5 (that is, by 10). That removes 5 from the running, at least temporarily.  Here is the progression:

1234560/2*5 = 123456
123456/2    = 61728
61728/2     = 30864
30864/2     = 15432
15432/2     = 7716
7716/2      = 3858
3858/2      = 1929
1929/3      = 543
543/3       = 271

My guess is that 271 is prime. It is one over 270, which has obvious factors.  Further investigation bears me out.

The "Five and Dime" Store.

Being that 5*2 is 10, and inversely 10/2 is 5, multiplying and dividing by 5 is easy. You only need to multiply by 10  and divide by 2 to get any number multiple of 5.

123*5 = 123*10/2 = 1230/2 = 615

Using the decimal point, it works the other way:

(123/2)(10) =61.5*10 = 615

Finally, it is easy to spot numbers divisible by 25 (5^2) and 125 (5^3). This is because 100 and 1000 are multiples of 10.

For 25, there are 3 multiples: 25, 50 and 75.  For example:

123,450 is divisible by 5 and 25. 50 is 2*5*5. So 2, 5, 10 and 25 are factored out.

123,450/2*5 = 12,345
12,345/5    = 6,171
6,171/3     = 2057, an odd number.  Is it prime? Stay tuned.

Another, example:

987,625 is divisible by 125 (5*25=625).  This would also place a 5 on the end of the next factor, raising the power by another 5.

8*125 = 2^3*5^2 = 10^2 = 1000

This leaves 987,000.  Removing the zeros, we then look for the factors, if any, of 987.  3 works, though 9 doesn't.

987/3 = 329.

So, dividing an odd by an odd, we've ended up with an odd number.  Further calculations are needed, but our odd number is within reach.  20*20 is 400, setting a limit.

So, what do I know?

I know that 5 and 10 are closely related by the prime number 2.  And this makes multiplying and dividing by 5 very easy. There are no other prime numbers related to 5, making the possibility of a random number being prime at less than 25%.

Even so, mathematically, there are an infinite number of prime numbers.  This is because "infinity" can not be divided. There are an infinite number of fractions.

Very odd, so to speak, that when not all odd numbers are prime, there can be an infinite number of primes anyway.

I know my brain hurts contemplating that.

To restate the FACTS:

1. The natural number 5 is prime.

2. 5 = 10/2

3. 5*2 = 10

4. Therefore, all numbers ending in 5 or 0 are multiples of 5.