Input a hex number as a string, convert to a list and any letters to uppercase.

Will plan to go from left to right, ie loop through the list of hex digits.

Initialize carry = 0

Since we are dividing by 11, carry can be any integer from 0 to 10.

We are interested in 16 times the carry plus the current digit, but we can make that nicer and equivalent mod 11 by being interested in 5 times the carry plus the current digit.  But we can make it even nicer by making a list with 11 entries of what will be the mod 11 contribution of the carry to the next carry.   That list is:

prior = [0,5,10,4,9,3,8,2,7,1,6]

And prior[carry] is the contribution to the next carry.

Also, let's make a dictionary where "key" is any hex digit and "value" is the decimal equivalent.

The next carry is the mod 11 value of (currentDigit plus prior[carry])

When there are no more digits, final carry value is the answer.

[If we were not even allowed to do the mod11 step, we could make a larger dictionary where the keys would be the concatenations of every possible carry with every possible current digit (11*16=176 entries) and the values the corresponding carry. I did not write the code for this part however] 

the following I believe will work on a hexadecimal integer of however many digits the programming language and machine constraints allow.

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prior = [0,5,10,4,9,3,8,2,7,1,6]

hexDigits = {}

for i in range(16):

    hexDigits[str(base2base(i,10,16))] = i

def r(str16):

    """ string16 is a string version of a base 16 number

    output the remainder if you divide by hex B  or dec 11  """

    carry = 0

    list16 = list(str16.upper())

    for d in list16:

        carry = (prior[carry] + hexDigits[d])%11

    if carry == 0:

        print(str16.upper(), 'is divisible by hex B.')

    return carry