📂Lemmas

Simplifying the Exponentiation of Two-digit Numbers Ending in 5

Formulas

The square of a two-digit number whose ones place is 5 can be computed quickly and easily as shown in the photo above. It’s okay to just know and use the result, but some might be curious about why it works this way.

Proof

Let’s assume any two-digit number whose ones place is $5$ is $10a+5$. Then, the square can be calculated as follows.

$$\begin{align*} (10a+5)(10a+5) =&\ 100a^2+100a+25 \\ =&\ 100a(a+1)+25 \end{align*}$$

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Explanation

Applying the above result, it’s easy to calculate even when the tens places are not the same.

If the tens place differs by 1, use $75$ instead of $25$, and do $+2$ instead of $+1$ for the multiplication of the tens place.

$$\begin{align*} (10a+5)\left[ 10(a+1)+5) \right] =&\100a(a+1)+100a+50+25 \\ =&\ 100a(a+2)+75 \end{align*}$$

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By now, you might think that it’s also possible to create a formula for when the difference between the tens places is $n$.

Multiplication of two two-digit numbers whose ones place is 5 and the difference in the tens place is $n$

$$\begin{align*} (10a+5)\left[ 10(a+n)+5 \right] =&\ 100a(a+n)+100a+50n+25 \\ =&\ 100a(a+n+1)+50n+25 \end{align*}$$

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For the tens digits, multiply (smaller number) by (larger number + 1), then multiply the result by 100, add 25, and also add 50 times the difference in the tens digits. It may seem complicated, but it’s actually not difficult once you try it. For the case of $25 \times 75$, since it’s $2 \times 8=16$, it would be $1600+250+25=1875$. There’s no need to memorize the general formula; knowing just the squares or situations where there’s a difference of 1 is enough for application.

Examples

$$25 \times 75=25(25+50)=25(25+25\times 2)=625+625\times 2=625\times 3=1875$$

Or

$$25 \times 75=100 \times 16 + 50 \times 5 +25=1600+250+25=1875$$