Quick Calculations 1: Squaring numbers divisible by 5

How can we quickly square numbers divisible by 5?

Lets go into an example: $145^2$ .

We split it into 2 parts : 14 and 5.

The last two digits are 25.

Then, the next succesive digits are $14 \cdot 15$ , or $14 \cdot (14+1)$ .

Therefore, $145^2=21025$ .

Why does this work?

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Comments

Simple. Take an $n$ digit number, where $n\ge2$ , as $10x+5$ . Thus, the number squared is $(10x+5)^{2}$ $=$ $100x^{2} + 100x + 25$ . Since the coefficients of $x^{2}$ and $x$ are multiplied by $100$ , the last two digits are $25$ . Now the next digits are given by $100(x)(x+1)$ , thus proving the property

Nanayaranaraknas Vahdam - 7 years, 1 month ago

@David Lee, I knew that!

Ameya Salankar - 7 years, 1 month ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$