## [Math] Factlets about integer factorization

There is a toyish *NIX utility factor available, which can factorize small numbers.

GCD is simply common factors. Here is 3*3:

% factor
9999999999
9999999999: 3 3 11 41 271 9091
87654321
87654321: 3 3 1997 4877


Let's check:

% python3
Python 3.8.10 (default, Jun 22 2022, 20:18:18)
[GCC 9.4.0] on linux
>>> import math
>>> math.gcd(9999999999, 87654321)
9
>>>


Obviously, prime numbers have no factors:

% factor
409993
409993: 409993


Hence, any number smaller than a prime will be coprime with it, because they don't share common factors. (But larger number can share.) GCD=1 for coprime numbers.

>>> math.gcd(409993, 10)
1
>>> math.gcd(409993, 100)
1
>>> math.gcd(409993, 123)
1
>>> math.gcd(409993, 999)
1


If you know common factors of x and y, you can do x/y division operation faster by dropping these common factors.

% factor
999999999
999999999: 3 3 3 3 37 333667
87654321
87654321: 3 3 1997 4877


Remove "3 3" from both lists of factors.

% bc
bc 1.07.1
Copyright 1991-1994, 1997, 1998, 2000, 2004, 2006, 2008, 2012-2017 Free Software Foundation, Inc.
This is free software with ABSOLUTELY NO WARRANTY.
For details type warranty'.

scale=5

999999999/87654321
11.40845

3*3*37*333667
111111111

1997*4877
9739369

111111111/9739369
11.40845

(3*3*37*333667)/(1997*4877)
11.40845
`

Division of smaller numbers is faster, of course.

###### (the post first published at 20220724.)

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