Brute force

The problem can be split into two parts:

• Check if the permutation satisfies the divisibility property.
• Find all the permutations of the digits \( 0 \) to \( 9 \).

The first part can be done by checking if a particular section of the number is divisible by the corresponding prime number.

From solution1.py:

def is_sub_string_divisible(n):
    divisors = [2, 3, 5, 7, 11, 13, 17]
    return all(int(str(n)[i + 1 : i + 4]) % d == 0 for i, d in enumerate(divisors))

The second part is done by using the itertools.permutations function. With the caveat that any permutation starting with \( 0 \) should be disregarded.

From solution1.py:

def sub_string_divisibility():
    res = 0
    for i in itertools.permutations("9876543210"):
        if i[0] == "0":
            continue
        x = int("".join(i))
        if is_sub_string_divisible(x):
            res += x
    return res

It is worth noting that if we encounter a permutation beginning with \( 0 \), we can immediately return the result without further iterations, as all other permutations will also start with \( 0 \).