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Codeforces Problem 1855B - Longest Divisors Interval

Posted by R.A.S CODERS
Code Intuition Competitive programming

Codeforces Problem 1855B - Longest Divisors Interval

Longest Divisors Interval – Exploring the Interconnectedness in Number Theory

problem link: Codeforces Problem 1855B - Longest Divisors Interval from Codeforces.

When working with numbers, it's fascinating how simple observations can reveal deep connections. The "Longest Divisors Interval" problem gives us a perfect opportunity to see how the structure of divisors affects the solution. This problem isn’t just about finding divisors—it’s about understanding how they are connected and how this connection shapes the solution.

The Problem Explained Simply

You’re given a positive integer n and asked to find the largest range of integers [l, r] where every integer in that range divides n. In other words, you need to find the longest interval of numbers such that all numbers in the range can divide n without leaving a remainder.

For example, if n = 6, the interval of divisors would be numbers like 1, 2, and 3, since 6 is divisible by each of them. The goal is to find how far this interval can extend before encountering a number that doesn’t divide n.

Key Insight: The Cascading Effect of Divisibility

Now, here’s the key point that will help us solve the problem efficiently:

If a number x does not divide n, then none of its multiples (like 2x, 3x, ...) will divide n either.

This is a very important observation because it helps us understand how divisors are connected. Let’s break it down with an example:

  • Suppose n = 6. The number 1 divides 6 (since 6 % 1 = 0), so 1 is part of our interval.
  • Next, we check 2. Since 6 % 2 = 0, 2 is also part of the interval.
  • We then check 3. Since 6 % 3 = 0, we include 3.
  • But when we check 4, we find that 6 % 4 != 0. This means 4 cannot divide 6, and none of the multiples of 4 (like 8, 12, etc.) can divide 6 either.

So, the moment we find that a number doesn't divide n, we know the interval must end just before that number. In our case, for n = 6, the largest valid interval is from 1 to 3.

Why Does This Matter?

This observation gives us a powerful shortcut: the length of the interval depends on the first number x where n % x != 0. Once we hit such an x, we know that the interval ends right before it. We don’t need to check each number individually, as the failure of one divisor immediately tells us the length of the interval.

Example Walkthroughs

Example 1: n = 6

  • Divisors of 6: 1, 2, 3, 6.
  • We start with 1, which divides 6, so we include it.
  • Next, we check 2. Since 6 % 2 = 0, we include 2.
  • We then check 3. Since 6 % 3 = 0, we include 3.
  • Finally, we check 4, and we find 6 % 4 != 0. So, we stop here. The longest interval is from 1 to 3, which has a length of 3.

Example 2: n = 10

  • Divisors of 10: 1, 2, 5, 10.
  • We start with 1, which divides 10.
  • Next, we check 2. Since 10 % 2 = 0, we include 2.
  • When we check 3, we find 10 % 3 != 0. So, we stop here. The longest interval is from 1 to 2, with a length of 2.

Example 3: n = 15

  • Divisors of 15: 1, 3, 5, 15.
  • We start with 1, which divides 15.
  • Next, we check 2, and we find 15 % 2 != 0. So, we stop here. The longest interval is just 1, with a length of 1.

Code Implementations

Below are the implementations in Java and C++.

JAVA


import java.util.Scanner;

public class LongestDivisorsInterval {
    public static void main(String[] args) {
        Scanner sc = new Scanner(System.in);
        int t = sc.nextInt();  // Number of test cases
        
        while (t-- > 0) {
            long n = sc.nextLong();
            long i = 1;
            
            while (i <= n && n % i == 0) {
                i++;
            }
            
            System.out.println(i - 1);
        }
        
        sc.close();
    }
}

Conclusion

This problem is a great example of how understanding the relationships between numbers can help us solve problems more efficiently. The key insight—that if n is not divisible by x, it’s also not divisible by any multiple of x—simplifies the problem drastically.

By recognizing that divisors form a connected system, we can avoid unnecessary checks and directly find the length of the longest valid interval. This connection between divisors is a beautiful part of number theory, and problems like these showcase just how powerful and elegant mathematical structures can be.

In this post, we focused on the logic and insights behind the problem, leaving the code implementation to you. The next time you’re faced with a divisor-related problem, remember the cascading effect: one non-divisor breaks the whole chain!

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