乘法数字根和乘法持久性

Question

I wrote the below code to figure out multiplicative digital root and multplicative persistence. I am clear about multiplicative digital root but have a question on multiplicative persistence (MP).

In the below code, there are two while loops and to find MP have put a counter incrementing everytime the multiplication happens.

MP differs when the line count += 1 was included in inner while loop and when kept in outer while loop.

Output when count += 1 is in the outer loop is 222 : [8, 3]

Now my question is which is correct for Muliplicative persistence.

Have referred https://rosettacode.org/wiki/Digital_root/Multiplicative_digital_root and as per this count += 1 is in the outer loop and the output is 222 : [8, 1]

Can anyone please suggest which is one is correct?

    def mdr_mp(num):
    '''
    This function computes the multiplicative digital root and multiplicative persistence of a given number
    '''
    product = 1
    mdr = num
    count = 0

    #Find mdr and persistence 
    while mdr > 9: #as long as mdr is > 9 
        while num > 0: #as long as quotient is > 0 
            #use divmod fn to return the quotient and remainder of num when divided by 10
            num, number = divmod(num, 10)
            product *= number #perform product of each digit in the number
            count += 1
        mdr = product #set product to mdr so as to check if it is > 9 else repeat
        num = product #set product to num so as to perform the product again
        product = 1 #initialize product to 1 so that product of the new num can be computed
        #count += 1 #multiplicative persistence 

    return [mdr, count] #returns multiplicative digital root and multiplicative persistence as a list

    num = 222
    #Function call returns list of number containing mdr and mp
    print("Number: (MDR, MP)")
    list1 = mdr_mp(num)
    print(num, ": ", list1)

Output:
    Number: (MDR, MP)
    222 :  [8, 3]

Answer 1

for 222, the (mdr, mp) is (8, 1) . -> 2*2*2 = 8 done in one loop.

for 34, the (mdr, mp) is (2, 2) . -> 3*4 = 12. 1*2 = 2. done in 2 loops.

programmatically:

def get_mdr_and_mp(number):
    time_multiplied = 0
    current_mdr = number
    while current_mdr > 9:
        current_mdr = multiply_all_digits(current_mdr)
        times_multiplied += 1
    return current_mdr, times_multiplied