The O(n) Club: When Prime Factors Make You a Copy-Paste Ninja (2 Keys Keyboard)

The O(n) Club: When Prime Factors Make You a Copy-Paste Ninja (2 Keys Keyboard)

⚡ TL;DR

You have two buttons and a dream: Copy All and Paste. Get exactly n ‘A’s with as few panic-clicks as possible. If you’re about to simulate every mindless combo, please—just factorize n first. The magic? Add up the prime factors of n.

// Actual Java wisdom
public int minSteps(int n) {
    int res = 0;
    for (int d = 2; d <= n; d++) {
        while (n % d == 0) {
            res += d;
            n /= d;
        }
    }
    return res;
}

🧠 How Devs Usually Mess This Up

Every newbie (and, let’s be honest, half the seniors on Friday afternoons) thinks “Just copy whenever you grow. Paste relentlessly!” But unless you’re coding on a typewriter, that’s extra work for zero payoff. Simulating every path? Your CPU will revolt, and your interviewer’s face will melt. Remember: only whole-clipboard copies count—none of that partial-copy sci-fi.

🔍 What’s Actually Going On

Imagine your problems are widgets. Got 1 widget, want 9? Don’t duplicate willy-nilly—factorize! Prime factors show where each mega-copy should happen: each factor is a boss level. Every time you find a divisor of n, you “copy,” and then you “paste” (factor – 1) times. One big copy is worth dozens of tiny ones; your clipboard becomes a turbocharger instead of a moped.

🛠️ PseudoCode

• Start with 1 A on screen.
• For i from 2 up to n:
• If n is divisible by i:
• Add i to your operation count.
• Divide n by i (since you just built i times as many).
• Repeat for the new n. Each division = one copy-paste segment.
• The process ends when n hits 1. The accumulated total is your golden number.

// java-ish pseudocode
steps = 0
for i = 2 to n:
   while n % i == 0:
      steps += i
      n /= i
return steps

💻 The Code

public int minSteps(int n) {
    int steps = 0;
    for (int factor = 2; factor <= n; factor++) {
        while (n % factor == 0) {
            steps += factor;
            n /= factor;
        }
    }
    return steps;
}

⚠️ Pitfalls, Traps & Runtime Smacks

• Brute force burns hot: Simulating every copy/paste not only costs time, but also your will to live. Watch out for O(n^2) soup.
• Clipboard rules: You always copy everything—no creative editing.
• Edge cases: n = 1? Zero steps, buddy; you already have your single ‘A’. n prime? It chugs along at n steps—one copy, lots of paste.
• Complexity: O(√n) time (that’s brisk!) and O(1) space. Java’s GC won’t even notice.

🧠 Interviewer Brain-Tattoo

“If your only plan is paste, you’re just making a mess. Factorize, then paste like a grown-up.”