Speedy Squares: Ends in 5!

Ever need to quickly square a number like 35 or 75 without a calculator? It's easier than you think! What if you could do it in seconds?

Subject: mental-calculation • Classes: 6–12 • Difficulty: intermediate

The Trick

To square any number ending in 5: 1. Ignore the '5' for a moment. Take the digit(s) before it. Let this number be $N$. 2. Multiply $N$ by the next consecutive integer, $(N+1)$. 3. Simply write '25' after the result from step 2. That's your answer! WHY IT WORKS: Any number ending in 5 can be written as $(10N + 5)$. Squaring this gives $(10N + 5)^2 = (10N)^2 + 2(10N)(5) + 5^2 = 100N^2 + 100N + 25 = 100N(N+1) + 25$. The $100N(N+1)$ part means you calculate $N(N+1)$ and then multiply by 100 (which shifts it to the left by two places), and then add 25 to the end. So, $N(N+1)$ forms the initial digits, and 25 is appended.

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