DIVISIBILITY CONJECTURE.

"""
EVIDENCE FOR A CONJECTURE IS PRESENTED.  THE CONJECTURE IS:

THE 3-DIVISIBILITY RULE IN BASE "10" GENERALIZES TO AN ARBITRARY BASE "Z" FOR ANY FACTOR OF "Z- 1".  NAMELY:

IF Y IS A FACTOR OF Z- 1, AND,
Y IS A FACTOR OF THE SUM OF THE DIGITS (OR PLACE-VALUE ENTRIES) OF A NUMBER X IN BASE Z, THEN,
Y IS A FACTOR OF X IN BASE Z.

SEE ALSO:
DIVISIBILITY RULE.  https://en.wikipedia.org/wiki/Divisibility_rule .
"""

import ctypes as CTYPES, random as RANDOM, string as STRING
CTYPES.windll.Kernel32.SetConsoleTitleW( "__DIVISIBILITY.PY__.__"+ "".join( RANDOM.choice( STRING.digits ) for _ in range( RANDOM.randint( 10000, 60000 ) ) ) )

for ITERATION in range( 99999999 ):
    import random as RANDOM
    RANDOM_SEED= RANDOM.randint( 0, 999999999 )
    RANDOM.seed( RANDOM_SEED )
    if ITERATION& 0XFFF== 0:
        print( "ITERATION:", ITERATION, ", RANDOM_SEED:", RANDOM_SEED, "." )
    Z= RANDOM.randint( 2, 29 )
    Y= [ RANDOM.randint( 0, Z- 1 ) for _ in range( RANDOM.randint( 1, 19 ) ) ]
    X= sum( Y[ W ]* Z** W for W in range( len( Y ) ) )
    W= sum( Y )
    for V in range( 2, Z ):
        if ( Z- 1 )% V:
            continue
        if ( Z- 1 )% V== 0 and W% V== 0:
            assert X% V== 0
        else:
            assert X% V!= 0