Prime Factors – Powers and Divisibility thereof.
There’s kind of an interesting thing that happens when you look into exactly what numbers are divisible by a given prime number.
It’s easiest to see with 2.
Every other number is divisible by 2.
2, 4, 6, 8, 10, 12, 14, 16, 18, 20 etcetera Divisible by 2
Every other number that is divisible by 2 is only divisible by 2 ONCE.
2, 6, 10, 14, 18, 22, 26, 30, 34, 38, 42, etcetera. Divisible by 2 and NOT by 4.
Every other number that is not divisible by 2 once is divisible by 2 TWICE, or rather, is divisible by 4.
4, 8, 12, 16, 20, 24, 28, 32, 36, 40, etcetera Divisible by 4
Every other number that is divisible by 4 is not divisible by 2 again.
4, 12, 20, 28, 36, 44, 52, 60, etcetera. Divisible by 4, but not by not 8.
It goes on like that, every other number that is divisible by a power of 2 is not divisible by any higher power of 2.
Every other number, every other 2, every other 4, every other 8, every other 16 … is divisible by that power of 2 and none higher.
And the same thing is true of higher numbers. Except instead of being every other, it’s every (that many) can be higher.
Every third number is divisible by 3(2/3 of all numbers are not divisible by 3)
2/3 of all 3s are divisible by 3 and not 9
2/3 of all 9s are divisible by 9 and not 27
2/3 of all 27s are divisible by 27 and not 81.
…
Every fifth number is divisible by 5(4/5 of all numbers are not divisible by 5)
4/5 of all 5s are divisible by 5 and not 25.
4/5 of all 25s are divisible by 25 and not 125.
4/5 of all 125s are divisible by 125 and not 625.
…
Every seventh number is divisible by 7(6/7 of all numbers are not divisible by 7)
6/7 of all 7s are divisible by 7 and not 49.
6/7 of all 49s are not divisible by 343…
and so on.
And combining all these rules together for every combination of primes, we also come up with every number.
I have no idea if, why, or how this is actually useful.
But you could also express all these numbers as a power of a prime number. With the first one being to the 0th power, which is one.
2/3 of all numbers divisible by 3^0 are not divisible by 3^1 🙂
2/3 of all numbers divisible by 3^1 are not divisible by 3^2 … and so on.
