🧮 Most Useful Modular Arithmetic Property to Simplify Calculations

Question: What is a simple modular arithmetic property you use most often to quickly simplify a calculation?

Modular arithmetic makes working with large numbers easier by focusing only on their remainders. Among its many properties, one stands out as the most practical and frequently used for quick simplifications.

🔹 The Core Property

The most useful property is:

(a + b) mod m = ((a mod m) + (b mod m)) mod m
(a × b) mod m = ((a mod m) × (b mod m)) mod m

This means you can take the remainder (mod) of numbers as you go — without having to calculate the full product or sum first.

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💡 Why It Works

Instead of handling large numbers directly, modular arithmetic lets you reduce them early. This keeps calculations smaller, faster, and error-free.

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⚙️ Example

Find (2347 × 6789) mod 10

(2347 mod 10) = 7
(6789 mod 10) = 9
(7 × 9) mod 10 = 63 mod 10 = 3

✅ Final Answer: 3

You simplified the whole problem just by working with remainders!

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🧠 More Quick Modular Tricks

• ab mod m = ((a mod m)b) mod m – great for large powers.
• (-a) mod m = m - (a mod m) – avoids negative remainders.
• If a ≡ b (mod m)