Binary numbers form the core foundation of computer science. As we’ve transitioned into a digital age, binary operations and their manipulations have become increasingly prevalent in coding challenges. One such interesting problem on Leetcode is the “Binary Prefix Divisible By 5”. This problem not only tests one’s understanding of binary numbers but also incorporates mathematical concepts. In this article, we’ll dissect this problem and present a step-by-step Python solution.

### Table of Contents

- Problem Statement
- Understanding Binary Numbers
- The Mathematics Behind the Problem
- Solution Strategy
- Python Implementation
- Time and Space Complexity Analysis
- Conclusion

### 1. Problem Statement

Given an array `A`

consisting of `N`

integers (either 0 or 1), interpret it as the binary representation of a number. For every prefix of the array, check if the number formed by that prefix is divisible by 5.

Return an array `answer`

where `answer[i]`

is `True`

if the first `(i+1)`

elements of `A`

are divisible by 5.

**Example:**

```
Input: A = [1,0,1]
Output: [False, False, True]
Explanation:
The numbers formed by the prefixes are:
1 -> Not divisible by 5
10 -> Not divisible by 5
101 -> Divisible by 5
```

### 2. Understanding Binary Numbers

Binary is a base-2 system, meaning it only has two digits: 0 and 1. When we read a binary number from left to right, each position represents an increasing power of 2.

For instance, the binary number `101`

in base 10 (decimal) is calculated as:

### 3. The Mathematics Behind the Problem

To efficiently check if each prefix of the array is divisible by 5, we don’t need to compute the whole number for each prefix. Instead, we can keep track of the current number and update it as we traverse the array.

Given a number `x`

, if you append a bit `b`

to its end in binary, the new number becomes `2x + b`

.

### 4. Solution Strategy

- Initialize
`current_num`

to 0. - Traverse through the array. For each bit:
- Update the
`current_num`

: current_num=2×current_num+bit - Check if
`current_num`

is divisible by 5. Append the result to the answer list. - Use modulo operation to avoid overflow: current_num=current_num%5

- Update the

### 5. Python Implementation

```
def prefixesDivBy5(A):
answer = []
current_num = 0
for bit in A:
# Update current number and take modulo to avoid overflow
current_num = (2 * current_num + bit) % 5
answer.append(current_num == 0)
return answer
```

### 6. Time and Space Complexity Analysis

**Time Complexity:**- We’re traversing the array once and performing constant-time operations for each element.
- Thus, the time complexity is O(n), where n is the length of the array.

**Space Complexity:**- We are using an extra array (
`answer`

) that has the same length as the input array. - Thus, the space complexity is O(n).

- We are using an extra array (

### 7. Conclusion

The “Binary Prefix Divisible By 5” problem on Leetcode elegantly fuses the realms of binary operations and mathematical insights. It serves as a brilliant example of how computer science problems often require a combination of domain knowledge and algorithmic thinking.