Python 如何找到最小的正整数使数字单调？

我发现了一个无法解决的编程问题。我得到了一组整数。对于

A

中的所有数字

x

，找到最小的正整数

y

，使

x*y

的数字增加或减少，并且乘积

x*y

是可能的最小值。例如，如果

A=（3637261089）

则

n=（184573137589182736455463728191）

给出数字

（6699999999999999999999999999）

但是有一些我的程序无法解决的难题。所有案例如下所示：

363726 1089 1313 1452 1717 1798 1815 1919 2121 2156 2178 2189 2541 2626 2805
2904 2997 3131 3267 3297 3434 3630 3838 3993 4037 4092 4107 4191 4242 4257 4312
4334 4343 4356 4378 4407 4532 4646 4719 4747 4807 4949 5011 5055 5071 5082 5151
5214 5353 5423 5445 5454 5495 5610 5665 5731 5808 5819 5858 5951 5989 5994 6171
6248 6281 6429 6446 6468 6523 6534 6565 6567 6594 6721 6767 6868 6897 6919 7051
7077 7128 7139 7171 7227 7260 7381 7424 7474 7513 7623 7678 7831 7858 7878 7881
7909 7986 8041 8063 8074 8088 8107 8129 8162 8173 8184 8195 8214 8283 8316 8349
8382 8415 8453 8484 8514 8624 8649 8712 8756 8778 8814 8932 8987 8989 8990 8991
9053 9064 9075 9099 9101 9119 9141 9156 9191 9213 9251 9292 9309 9328 9361 9393
9438 9493 9515 9546 9595 9597 9603 9614 9667 9678 9757 9797 9801 9802 9834 9890
9898 9909

这是我的慢程序：

def find_smallest_increasing(number, length):
    ehd = -1
    num = "0"
    length += 1
    for one in range(0,length):
        for two in range(0,length-one):
            for three in range(0,length-one-two):
                for four in range(0,length-one-two-three):
                    for five in range(0,length-one-two-three-four):
                        for six in range(0,length-one-two-three-four-five):
                            for seven in range(0,length-one-two-three-four-five-six):
                                for eight in range(0,length-one-two-three-four-five-six-seven):
                                    for nine in range(0,length-one-two-three-four-five-six-seven-eight):
                                        if max(one,two,three,four,five,six,seven,eight,nine) > 0:
                                            num = "1"*one+"2"*two+"3"*three+"4"*four+"5"*five+"6"*six+"7"*seven+"8"*eight+"9"*nine
                                            if int(num) % number == 0:
                                                if ehd == -1:
                                                    ehd = int(num)
                                                if int(num) < ehd:
                                                    ehd = int(num)
    return(ehd)

def find_smallest_decreasing(number, length):
    ehd = -1
    num = "0"
    length += 1
    for one in range(0,length):
        for two in range(0,length-one):
            for three in range(0,length-one-two):
                for four in range(0,length-one-two-three):
                    for five in range(0,length-one-two-three-four):
                        for six in range(0,length-one-two-three-four-five):
                            for seven in range(0,length-one-two-three-four-five-six):
                                for eight in range(0,length-one-two-three-four-five-six-seven):
                                    for nine in range(0,length-one-two-three-four-five-six-seven-eight):
                                        for zero in range(0,length-one-two-three-four-five-six-seven-eight-nine):
                                            if max(one,two,three,four,five,six,seven,eight,nine) > 0:
                                                num = "9"*one+"8"*two+"7"*three+"6"*four+"5"*five+"4"*six+"3"*seven+"2"*eight+"1"*nine+"0"*zero
                                                if int(num) % number == 0:
                                                    if ehd == -1:
                                                        ehd = int(num)
                                                    if int(num) < ehd:
                                                        ehd = int(num)
    return(ehd)

numbers = [363,726,1089, 1313, 1452, 1717, 1798, 1815, 1919, 2121, 2156, 2178, 2189, 2541, 2626, 2805,
2904, 2997, 3131, 3267, 3297, 3434, 3630, 3838, 3993, 4037, 4092, 4107, 4191, 4242, 4257, 4312,
4334, 4343, 4356, 4378, 4407, 4532, 4646, 4719, 4747, 4807, 4949, 5011, 5055, 5071, 5082, 5151,
5214, 5353, 5423, 5445, 5454, 5495, 5610, 5665, 5731, 5808, 5819, 5858, 5951, 5989, 5994, 6171,
6248, 6281, 6429, 6446, 6468, 6523, 6534, 6565, 6567, 6594, 6721, 6767, 6868, 6897, 6919, 7051,
7077, 7128, 7139, 7171, 7227, 7260, 7381, 7424, 7474, 7513, 7623, 7678, 7831, 7858, 7878, 7881,
7909, 7986, 8041, 8063, 8074, 8088, 8107, 8129, 8162, 8173, 8184, 8195, 8214, 8283, 8316, 8349,
8382, 8415, 8453, 8484, 8514, 8624, 8649, 8712, 8756, 8778, 8814, 8932, 8987, 8989, 8990, 8991,
9053, 9064, 9075, 9099, 9101, 9119, 9141, 9156, 9191, 9213, 9251, 9292, 9309, 9328, 9361, 9393,
9438, 9493, 9515, 9546, 9595, 9597, 9603, 9614, 9667, 9678, 9757, 9797, 9801, 9802, 9834, 9890,
9898, 9909]

for k in range(0,len(numbers)):
    number = numbers[k]
    a = -1
    b = -1
    i= 1
    j= 1
    while a == -1:
        if a % 10 != 0:
            a = find_smallest_increasing(number,i)
        else:
            a = -1
        i = i + 1
    while b == -1:
        b = find_smallest_decreasing(number,max(i,j))
        j = j + 1
    print(str(number)+" "+str(min(a,b)/number)+" " + str(min(a,b)))

第二次尝试：

def generate_all_numbers(length):
    l = list()
    for one in range(0,length):
        for two in range(0,length-one):
            for three in range(0,length-one-two):
                for four in range(0,length-one-two-three):
                    for five in range(0,length-one-two-three-four):
                        for six in range(0,length-one-two-three-four-five):
                            for seven in range(0,length-one-two-three-four-five-six):
                                for eight in range(0,length-one-two-three-four-five-six-seven):
                                    for nine in range(0,length-one-two-three-four-five-six-seven-eight):
                                        for ten in range(0,length-one-two-three-four-five-six-seven-eight-nine):
                                            if max(one,two,three,four,five,six,seven,eight,nine) > 0:
                                                num1 = "1"*one+"2"*two+"3"*three+"4"*four+"5"*five+"6"*six+"7"*seven+"8"*eight+"9"*nine
                                                num2 = "9"*one+"8"*two+"7"*three+"6"*four+"5"*five+"4"*six+"3"*seven+"2"*eight+"1"*nine+"0"*ten
                                                l.append(int(num1)) 
                                                l.append(int(num2))
    return(list(set(l)))

numbers = [363,726,1089, 1313, 1452, 1717, 1798, 1815, 1919, 2121, 2156, 2178, 2189, 2541, 2626, 2805,
2904, 2997, 3131, 3267, 3297, 3434, 3630, 3838, 3993, 4037, 4092, 4107, 4191, 4242, 4257, 4312,
4334, 4343, 4356, 4378, 4407, 4532, 4646, 4719, 4747, 4807, 4949, 5011, 5055, 5071, 5082, 5151,
5214, 5353, 5423, 5445, 5454, 5495, 5610, 5665, 5731, 5808, 5819, 5858, 5951, 5989, 5994, 6171,
6248, 6281, 6429, 6446, 6468, 6523, 6534, 6565, 6567, 6594, 6721, 6767, 6868, 6897, 6919, 7051,
7077, 7128, 7139, 7171, 7227, 7260, 7381, 7424, 7474, 7513, 7623, 7678, 7831, 7858, 7878, 7881,
7909, 7986, 8041, 8063, 8074, 8088, 8107, 8129, 8162, 8173, 8184, 8195, 8214, 8283, 8316, 8349,
8382, 8415, 8453, 8484, 8514, 8624, 8649, 8712, 8756, 8778, 8814, 8932, 8987, 8989, 8990, 8991,
9053, 9064, 9075, 9099, 9101, 9119, 9141, 9156, 9191, 9213, 9251, 9292, 9309, 9328, 9361, 9393,
9438, 9493, 9515, 9546, 9595, 9597, 9603, 9614, 9667, 9678, 9757, 9797, 9801, 9802, 9834, 9890,
9898, 9909]

l = generate_all_numbers(20)
A = list()
for i in range(len(l)):
    for j in range(len(numbers)):
        if l[i] % numbers[j] == 0:
             A.append(l[i])
B = list()
for j in range(len(numbers)):
 best = int("9" * 20)
 for i in range(len(A)):
    if A[i] % numbers[j] == 0:
        if A[i] < best:
            best = A[i]
 print(str(numbers[j])+" "+str(best/numbers[j])+ " " + str(best))

第三次尝试：我发现如果我将简单和困难的案例分开，我可以解决更多的案例：

def generate_all_numbers(length):
    l = list()
    for one in range(0,length):
        for two in range(0,length-one):
            for three in range(0,length-one-two):
                for four in range(0,length-one-two-three):
                    for five in range(0,length-one-two-three-four):
                        for six in range(0,length-one-two-three-four-five):
                            for seven in range(0,length-one-two-three-four-five-six):
                                for eight in range(0,length-one-two-three-four-five-six-seven):
                                    for nine in range(0,length-one-two-three-four-five-six-seven-eight):
                                        for ten in range(0,length-one-two-three-four-five-six-seven-eight-nine):
                                            if max(one,two,three,four,five,six,seven,eight,nine) > 0:
                                                num1 = "1"*one+"2"*two+"3"*three+"4"*four+"5"*five+"6"*six+"7"*seven+"8"*eight+"9"*nine
                                                num2 = "9"*one+"8"*two+"7"*three+"6"*four+"5"*five+"4"*six+"3"*seven+"2"*eight+"1"*nine+"0"*ten
                                                l.append(int(num1)) 
                                                l.append(int(num2))
    return(list(set(l)))

def find_smallest_increasing(number, length):
    ehd = -1
    num = "0"
    length += 1
    for one in range(0,length):
        for two in range(0,length-one):
            for three in range(0,length-one-two):
                for four in range(0,length-one-two-three):
                    for five in range(0,length-one-two-three-four):
                        for six in range(0,length-one-two-three-four-five):
                            for seven in range(0,length-one-two-three-four-five-six):
                                for eight in range(0,length-one-two-three-four-five-six-seven):
                                    for nine in range(0,length-one-two-three-four-five-six-seven-eight):
                                        if max(one,two,three,four,five,six,seven,eight,nine) > 0:
                                            num = "1"*one+"2"*two+"3"*three+"4"*four+"5"*five+"6"*six+"7"*seven+"8"*eight+"9"*nine
                                            if int(num) % number == 0:
                                                if ehd == -1:
                                                    ehd = int(num)
                                                if int(num) < ehd:
                                                    ehd = int(num)
    return(ehd)

def find_smallest_decreasing(number, length):
    ehd = -1
    num = "0"
    length += 1
    for one in range(0,length):
        for two in range(0,length-one):
            for three in range(0,length-one-two):
                for four in range(0,length-one-two-three):
                    for five in range(0,length-one-two-three-four):
                        for six in range(0,length-one-two-three-four-five):
                            for seven in range(0,length-one-two-three-four-five-six):
                                for eight in range(0,length-one-two-three-four-five-six-seven):
                                    for nine in range(0,length-one-two-three-four-five-six-seven-eight):
                                        for zero in range(0,length-one-two-three-four-five-six-seven-eight-nine):
                                            if max(one,two,three,four,five,six,seven,eight,nine) > 0:
                                                num = "9"*one+"8"*two+"7"*three+"6"*four+"5"*five+"4"*six+"3"*seven+"2"*eight+"1"*nine+"0"*zero
                                                if int(num) % number == 0:
                                                    if ehd == -1:
                                                        ehd = int(num)
                                                    if int(num) < ehd:
                                                        ehd = int(num)
    return(ehd)

numbers = [363,726, 1313, 1452, 1717, 1798, 1815, 1919, 2121, 2156, 2189, 2541, 2626, 2805,
2904, 2997, 3131, 3297, 3434, 3630, 3838, 3993, 4037, 4092, 4107, 4191, 4242, 4257, 4312,
4334, 4343, 4378, 4407, 4532, 4646, 4719, 4747, 4807, 4949, 5011, 5055, 5071, 5082, 5151,
5214, 5353, 5423, 5454, 5495, 5610, 5665, 5731, 5808, 5819, 5858, 5951, 5989, 5994, 6171,
6248, 6281, 6429, 6446, 6468, 6523, 6565, 6567, 6594, 6721, 6767, 6868, 6897, 6919, 7051,
7077, 7128, 7139, 7171, 7227, 7260, 7381, 7424, 7474, 7513, 7678, 7831, 7858, 7878, 7881,
7909, 7986, 8041, 8063, 8074, 8088, 8107, 8129, 8162, 8173, 8184, 8195, 8214, 8283, 8316, 8349,
8382, 8415, 8453, 8484, 8514, 8624, 8649, 8756, 8778, 8814, 8932, 8987, 8989, 8990, 8991,
9053, 9064, 9075, 9099, 9101, 9119, 9141, 9156, 9191, 9213, 9251, 9292, 9309, 9328, 9361, 9393,
9438, 9493, 9515, 9546, 9595, 9597, 9603, 9614, 9667, 9678, 9757, 9797, 9802, 9834, 9890,
9898, 9909]

hardnumbers = [1089, 2178, 3267, 4356, 5445, 6534, 7623, 8712, 9801]

l = generate_all_numbers(20)
A = list()
for i in range(len(l)):
    for j in range(len(numbers)):
        if l[i] % numbers[j] == 0:
             A.append(l[i])
B = list()
for j in range(len(numbers)):
 best = int("9" * 2000)
 for i in range(len(A)):
    if A[i] % numbers[j] == 0:
        if A[i] < best:
            best = A[i]
 print(str(numbers[j])+" "+str(best/numbers[j])+ " " + str(best))

for k in range(0,len(hardnumbers)):
    number = hardnumbers[k]
    a = -1
    b = -1
    i= 1
    j= 1
    while a == -1:
        if a % 5 != 0:
            a = find_smallest_increasing(number,i)
        i = i + 1
    b = -1
    j = 1
    while b == -1:
        b = find_smallest_decreasing(number,max(i,j))
        j = j + 1
    print(str(number)+" "+str(min(a,b)/number)+" " + str(min(a,b)))

def生成所有数字（长度）：
l=列表（）
对于范围（0，长度）中的一个：
对于范围内的两个（0，长度为1）：
对于范围内的三个（0，长度1-2）：
对于范围内的四个（0，长度一二三）：
对于范围内的五（0，长度一二三四）：
对于范围内的六（0，长度一二三四五）：
对于范围内的七（0，长度一二三四五六）：
对于范围内的八（0，长度一二三四五六七）：
范围为九（0，长度为一二三四五六七八）：
对于范围内的十（0，长度一二三四五六七八九）：
如果最大值（一、二、三、四、五、六、七、八、九）>0：
num1=“1”*1+“2”*2+“3”*3+“4”*4+“5”*5+“6”*6+“7”*7+“8”*8+“9”*9
num2=“9”*1+“8”*2+“7”*3+“6”*4+“5”*5+“4”*6+“3”*7+“2”*8+“1”*9+“0”*10
l、 附加（int（num1））
l、 附加（int（num2））
返回（列表（集合（l）））
def find_最小值_递增（数字、长度）：
ehd=-1
num=“0”
长度+=1
对于范围（0，长度）中的一个：
对于范围内的两个（0，长度为1）：
对于范围内的三个（0，长度1-2）：
对于范围内的四个（0，长度一二三）：
对于范围内的五（0，长度一二三四）：
对于范围内的六（0，长度一二三四五）：
对于范围内的七（0，长度一二三四五六）：
对于范围内的八（0，长度一二三四五六七）：
范围为九（0，长度为一二三四五六七八）：
如果最大值（一、二、三、四、五、六、七、八、九）>0：
num=“1”*1+“2”*2+“3”*3+“4”*4+“5”*5+“6”*6+“7”*7+“8”*8+“9”*9
如果int（num）%number==0：
如果ehd==-1：
ehd=int（num）
如果int（num）0：
num=“9”*1+“8”*2+“7”*3+“6”*4+“5”*5+“4”*6+“3”*7+“2”*8+“1”*9+“0”*0
如果int（num）%number==0：
如果ehd==-1：
ehd=int（num）
如果int（num）def generate_all_numbers(length):
    l = list()
    for one in range(0,length):
        for two in range(0,length-one):
            for three in range(0,length-one-two):
                for four in range(0,length-one-two-three):
                    for five in range(0,length-one-two-three-four):
                        for six in range(0,length-one-two-three-four-five):
                            for seven in range(0,length-one-two-three-four-five-six):
                                for eight in range(0,length-one-two-three-four-five-six-seven):
                                    for nine in range(0,length-one-two-three-four-five-six-seven-eight):
                                        for ten in range(0,length-one-two-three-four-five-six-seven-eight-nine):
                                            if max(one,two,three,four,five,six,seven,eight,nine) > 0:
                                                num1 = "1"*one+"2"*two+"3"*three+"4"*four+"5"*five+"6"*six+"7"*seven+"8"*eight+"9"*nine
                                                num2 = "9"*one+"8"*two+"7"*three+"6"*four+"5"*five+"4"*six+"3"*seven+"2"*eight+"1"*nine+"0"*ten
                                                l.append(int(num1)) 
                                                l.append(int(num2))
    return(list(set(l)))

def find_smallest_increasing(number, length):
    ehd = -1
    num = "0"
    length += 1
    for one in range(0,length):
        for two in range(0,length-one):
            for three in range(0,length-one-two):
                for four in range(0,length-one-two-three):
                    for five in range(0,length-one-two-three-four):
                        for six in range(0,length-one-two-three-four-five):
                            for seven in range(0,length-one-two-three-four-five-six):
                                for eight in range(0,length-one-two-three-four-five-six-seven):
                                    for nine in range(0,length-one-two-three-four-five-six-seven-eight):
                                        if max(one,two,three,four,five,six,seven,eight,nine) > 0:
                                            num = "1"*one+"2"*two+"3"*three+"4"*four+"5"*five+"6"*six+"7"*seven+"8"*eight+"9"*nine
                                            if int(num) % number == 0:
                                                if ehd == -1:
                                                    ehd = int(num)
                                                if int(num) < ehd:
                                                    ehd = int(num)
    return(ehd)

def find_smallest_decreasing(number, length):
    ehd = -1
    num = "0"
    length += 1
    for one in range(0,length):
        for two in range(0,length-one):
            for three in range(0,length-one-two):
                for four in range(0,length-one-two-three):
                    for five in range(0,length-one-two-three-four):
                        for six in range(0,length-one-two-three-four-five):
                            for seven in range(0,length-one-two-three-four-five-six):
                                for eight in range(0,length-one-two-three-four-five-six-seven):
                                    for nine in range(0,length-one-two-three-four-five-six-seven-eight):
                                        for zero in range(0,length-one-two-three-four-five-six-seven-eight-nine):
                                            if max(one,two,three,four,five,six,seven,eight,nine) > 0:
                                                num = "9"*one+"8"*two+"7"*three+"6"*four+"5"*five+"4"*six+"3"*seven+"2"*eight+"1"*nine+"0"*zero
                                                if int(num) % number == 0:
                                                    if ehd == -1:
                                                        ehd = int(num)
                                                    if int(num) < ehd:
                                                        ehd = int(num)
    return(ehd)

numbers = [363,726, 1313, 1452, 1717, 1798, 1815, 1919, 2121, 2156, 2189, 2541, 2626, 2805,
2904, 2997, 3131, 3297, 3434, 3630, 3838, 3993, 4037, 4092, 4107, 4191, 4242, 4257, 4312,
4334, 4343, 4378, 4407, 4532, 4646, 4719, 4747, 4807, 4949, 5011, 5055, 5071, 5082, 5151,
5214, 5353, 5423, 5454, 5495, 5610, 5665, 5731, 5808, 5819, 5858, 5951, 5989, 5994, 6171,
6248, 6281, 6429, 6446, 6468, 6523, 6565, 6567, 6594, 6721, 6767, 6868, 6897, 6919, 7051,
7077, 7128, 7139, 7171, 7227, 7260, 7381, 7424, 7474, 7513, 7678, 7831, 7858, 7878, 7881,
7909, 7986, 8041, 8063, 8074, 8088, 8107, 8129, 8162, 8173, 8184, 8195, 8214, 8283, 8316, 8349,
8382, 8415, 8453, 8484, 8514, 8624, 8649, 8756, 8778, 8814, 8932, 8987, 8989, 8990, 8991,
9053, 9064, 9075, 9099, 9101, 9119, 9141, 9156, 9191, 9213, 9251, 9292, 9309, 9328, 9361, 9393,
9438, 9493, 9515, 9546, 9595, 9597, 9603, 9614, 9667, 9678, 9757, 9797, 9802, 9834, 9890,
9898, 9909]

hardnumbers = [1089, 2178, 3267, 4356, 5445, 6534, 7623, 8712, 9801]

l = generate_all_numbers(20)
A = list()
for i in range(len(l)):
    for j in range(len(numbers)):
        if l[i] % numbers[j] == 0:
             A.append(l[i])
B = list()
for j in range(len(numbers)):
 best = int("9" * 2000)
 for i in range(len(A)):
    if A[i] % numbers[j] == 0:
        if A[i] < best:
            best = A[i]
 print(str(numbers[j])+" "+str(best/numbers[j])+ " " + str(best))

for k in range(0,len(hardnumbers)):
    number = hardnumbers[k]
    a = -1
    b = -1
    i= 1
    j= 1
    while a == -1:
        if a % 5 != 0:
            a = find_smallest_increasing(number,i)
        i = i + 1
    b = -1
    j = 1
    while b == -1:
        b = find_smallest_decreasing(number,max(i,j))
        j = j + 1
    print(str(number)+" "+str(min(a,b)/number)+" " + str(min(a,b)))

def nextMonotoneDigits(self, N):
        if N < 10: return N
        n, inv_index = N, -1
        num = [int(d) for d in str(n)[::-1]] 

        for i in range(1, len(num)): 
            if num[i] > num[i - 1] or (inv_index != -1 and num[inv_index] == num[i]):
                inv_index = i

        if inv_index == -1: return N

        for i in range(inv_index): num[i] = 9
        num[inv_index] -= 1

        return int(''.join([ str(i) for i in num[::-1]])) 

y = b_n*10^n + b_(n-1)*10^(n-1) ... + b_0*10^0

3 * 363 = 1089

b_1 * 3 + 8 = 9 (mod 10)
b_1 * 3 = 1 (mod 10)
b_1 = (10x + 1) / 3
b_1 = 7