Count Binary 1s Fast — No Full Conversion Needed!"

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If you’ve ever dealt with binary numbers—whether in competitive exams, technical interviews, or programming challenges—you know how important efficiency is. A common task is determining how many 1s appear in the binary representation of a number.

Of course, you could calculate the entire value, convert it to binary, and count the 1s. But that can be time-consuming. Fortunately, there’s a faster, smarter way to do it.

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🧠 The Problem: (7 × 256) + (4 × 16) + (9 × 4096) + 5

The usual way:

1. Multiply all the numbers

2. Add them up

3. Convert to binary

4. Count the 1s

It works, but it's slow. Let's do better.

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⚡ See the Pattern

First, rewrite the powers:

• 256 = 2⁸

• 16 = 2⁴

• 4096 = 2¹²

Now the expression becomes: (7 × 2⁸) + (4 × 2⁴) + (9 × 2¹²) + 5

Now just look at the small numbers:

• 7 → 111 → 3 ones

• 4 → 100 → 1 one

• 9 → 1001 → 2 ones

• 5 → 101 → 2 ones

Multiply by powers of 2? That just shifts the bits left — it doesn’t change how many 1s there are.

So total 1s = 3 + 1 + 2 + 2 = 8

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✅ Quick Method

1. Rewrite each term with powers of 2

2. Convert the small numbers to binary

3. Count the 1s in each one

4. Ignore bit shifting — doesn't change the 1 count

5. Add everything up

Done!

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🔍 Try This Example : (15 × 512) + 3

• 512 = 2⁹

• 15 → 1111 → 4 ones

• 3 → 11 → 2 ones

Answer: 4 + 2 = 6 ones

Quick and easy!

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💡 Why This Works

Multiplying by 2ⁿ just adds zeros to the end of the binary number. It shifts the digits left, but doesn’t add or remove any 1s.

So instead of converting large numbers to binary, just focus on the coefficient — the smaller number being multiplied.

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📌 Why It’s Useful

• Saves time in competitive exams

• No full binary conversion

• Improves binary pattern recognition

• Great for digital logic design

• Useful in programming interviews

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⚠️ Things to Remember

• Works best with (small number × power of 2)

• You should be able to convert small numbers (up to 31) into binary quickly

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🔧 Use Cases

• Digital circuit design

• Bit manipulation in code

• Hamming weight/parity calculations

• Competitive programming

• Technical interviews

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Final Thoughts

This binary shortcut is perfect for anyone looking to optimize their problem-solving speed. It’s fast, reliable, and a great way to build your confidence with binary math.

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