Step by Step solution about How to resolve the algorithm Minimum positive multiple in base 10 using only 0 and 1 step by step in the Python programming language

The Python code you provided is an implementation of the A004290 sequence, which is defined as the smallest number that is divisible by all numbers from 1 to n. This code uses a dynamic programming approach to compute the sequence efficiently.

The function getA004290 takes a single argument n, which is the number for which we want to compute the A004290 value.

The function first checks if n is less than 2. If it is, then the function simply returns 1, since the A004290 value for n < 2 is always 1.

If n is greater than or equal to 2, then the function creates a 2D array arr with dimensions n x n. The element arr[i][j] represents the smallest number that is divisible by all numbers from 1 to i and leaves a remainder of j when divided by n.

The function then initializes the first two rows of arr. The element arr[0][0] is set to 1, since the smallest number that is divisible by all numbers from 1 to 0 is 1. The element arr[0][1] is also set to 1, since the smallest number that is divisible by all numbers from 1 to 0 and leaves a remainder of 1 when divided by n is 1.

The function then enters a while loop that continues until the element arr[m - 1][-10 ** m % n] is equal to 1. This condition checks if the smallest number that is divisible by all numbers from 1 to m - 1 and leaves a remainder of -10 ** m % n when divided by n is equal to 1. If this condition is true, then the function has found the A004290 value for n, and the loop terminates.

Inside the loop, the function first sets the element arr[m][0] to 1. This is because the smallest number that is divisible by all numbers from 1 to m and leaves a remainder of 0 when divided by n is always 1.

The function then iterates over the elements of the mth row of arr, starting from the second element. For each element arr[m][k], the function sets it to the maximum of the two elements arr[m - 1][k] and arr[m - 1][k - 10 ** m % n]. This is because the smallest number that is divisible by all numbers from 1 to m and leaves a remainder of k when divided by n is either the smallest number that is divisible by all numbers from 1 to m - 1 and leaves a remainder of k when divided by n, or the smallest number that is divisible by all numbers from 1 to m - 1 and leaves a remainder of k - 10 ** m % n when divided by n.

After the loop has terminated, the function sets the variable r to 10 ** m. This is because the A004290 value for n is the smallest number that is divisible by all numbers from 1 to m, and this number is 10 ** m.

The function then sets the variable k to -r % n. This is because the A004290 value for n is the smallest number that is divisible by all numbers from 1 to n, and this number leaves a remainder of -r % n when divided by n.

The function then iterates over the elements of arr from row m - 1 to row 1, inclusive. For each element arr[j][k], the function checks if it is equal to 0. If it is, then the function sets r to r + 10 ** j and sets k to (k - 10 ** j) % n. This is because the A004290 value for n is the smallest number that is divisible by all numbers from 1 to n, and this number leaves a remainder of k when divided by n. However, if the element arr[j][k] is equal to 0, then this means that there is no number that is divisible by all numbers from 1 to j and leaves a remainder of k when divided by n. Therefore, the function must increase r by 10 ** j and decrease k by 10 ** j so that it can continue searching for the A004290 value for n.

Finally, the function checks if k is equal to 1. If it is, then the function sets r to r + 1. This is because the A004290 value for n is the smallest number that is divisible by all numbers from 1 to n, and this number leaves a remainder of k when divided by n. However, if k is equal to 1, then this means that the function has found the smallest number that is divisible by all numbers from 1 to n, and this number leaves a remainder of 1 when divided by n. Therefore, the function must increase r by 1 so that it returns the smallest number that is divisible by all numbers from 1 to n.

The function then returns the value of r, which is the A004290 value for n.

The code then tests the function for n values from 1 to 10, from 95 to 106, and for the values 297, 576, 594, 891, 909, 999, 1998, 2079, 2251, 2277, 2439, 2997, and 4878. For each n value, the code prints the A004290 value for n and the factorization of the A004290 value for n.

def getA004290(n):
    if n < 2:
        return 1
    arr = [[0 for _ in range(n)] for _ in range(n)]
    arr[0][0] = 1
    arr[0][1] = 1
    m = 0
    while True:
        m += 1
        if arr[m - 1][-10 ** m % n] == 1:
            break
        arr[m][0] = 1
        for k in range(1, n):
            arr[m][k] = max([arr[m - 1][k], arr[m - 1][k - 10 ** m % n]])
    r = 10 ** m
    k = -r % n
    for j in range((m - 1), 0, -1):
        if arr[j - 1][k] == 0:
            r = r + 10 ** j
            k = (k - 10 ** j) % n   
    if k == 1:
        r += 1
    return r
 
for n in [i for i in range(1, 11)] + \
          [i for i in range(95, 106)] + \
          [297, 576, 594, 891, 909, 999, 1998, 2079, 2251, 2277, 2439, 2997, 4878]:
    result = getA004290(n)
    print(f"A004290({n}) = {result} = {n} * {result // n})")