Minimum Factorization

Minimum Factorization

1. What is this problem about?

The Minimum Factorization problem asks you to find the smallest positive integer $n$ whose digits, when multiplied together, equal a given positive integer $a$. If no such $n$ exists or if the result exceeds the capacity of a 32-bit signed integer, you should return 0. This problem is about reversing the process of digit multiplication to find the most compact number possible.

2. Why is this asked in interviews?

Tencent and other companies ask the Minimum Factorization interview question to test a candidate's grasp of Greedy algorithms and basic number theory. It requires the insight that to make a number small, you should use as few digits as possible, which means using the largest possible digits (factors) first. It also tests your ability to handle integer overflow constraints effectively.

3. Algorithmic pattern used

The pattern is a Greedy approach. To minimize the final number, we want the fewest number of digits. To get the fewest digits, we should use the largest possible single-digit factors (9 down to 2). We repeatedly divide the input $a$ by the largest possible digit factor. The digits found are then placed in the units, tens, and hundreds places in reverse order (smaller digits at the front) to ensure the resulting number is the smallest. This "Math, Greedy interview pattern" is a clever way to solve factorization problems.

4. Example explanation

Suppose $a = 48$.

1. Can we divide by 9? No.
2. Can we divide by 8? Yes. $48 / 8 = 6$. Digit found: 8.
3. Can we divide 6 by 8, 7? No.
4. Can we divide 6 by 6? Yes. $6 / 6 = 1$. Digit found: 6.
5. Our digits are 8 and 6. To make the smallest number, we arrange them as 68. If $a = 15$, we get digits 5 and 3, result 35.

5. Common mistakes candidates make

In the Minimum Factorization coding problem, a frequent mistake is not checking for the 32-bit integer limit. The resulting number can easily exceed $2^{31}-1$, so you must use a 64-bit integer for intermediate calculations and check the range before returning. Another mistake is trying to find prime factors first; while related, you specifically need single-digit factors (which include composite numbers like 8 and 9) to minimize the number of digits.

6. Interview preparation tip

When trying to find the "smallest number" made of digits, always remember: fewer digits are better than smaller digits, and once the number of digits is fixed, smaller digits should be in the higher significance positions. This "Greedy digit placement pattern" is useful for many problems involving number construction.