in which base is the number 221 the factor of 1215

Question

I basically tried to find a number $n$ which would be the base and then divide both the numbers $221$ and $1215$ so as to get a combo which has same remainder But I am not able to proceed in doing so. Help would be appreciated

Answer 1

$$ 2 \left( x^{3} + 2 x^{2} + x + 5 \right) = \left( 2 x^{2} + 2 x + 1 \right) \cdot \left( x + 1 \right) - x + 9 $$

so try base $9$

We have $$ 221_{nine} = 181 $$ $$ 1215_{nine} = 905 $$ As both are odd, it did no damage when I multiplied the larger number by $2$

Answer 2

I wrote small python script to test all of the relevant bases

# what base is 221 a factor of 1215

snumber="221"
sother="1215"

def todecimal(input,base):
    output=0
    for i in range(len(input)):
        output=output+int(input[i])*(base**(len(input)-1-i))
    return output

for i in range(2,300):#starting at five is better cause largest number given is five,but 2 can be used anywhere if you want to test for other numbers.
    if((todecimal(sother,i)%todecimal(snumber,i))==0):
        print(i)

Its basically brute forcing it for the first 300 bases but you can test for any 2 numbers. Just change the input, dont need to import anything , any python 3 ide will be able to run this.