Math CentralQuandaries & Queries


Question from Aarru,

Sorry this may be a dumb question but how do you find possible unit digits of a
perfect square


This is not a dumb question.

I would start by writing the squares of the ten digits 0, 1, 2, ... , 9. What is the units digit of each square? Did you get every possible digit? Can you get a different digit if you square a larger integer?

Suppose $n$ is an integer and $d$ is its units digit. There is an integer $k$ so that

\[n = k \times 10 + a.\]

For example $165976 = 16597 \times 10 + 6.$

The square of $n$ can then be written

\[n^2 = \left(k \times 10 + d\right)^{2}.\]

Expanding the right side and simplifying I get

\[n^2 = k^2 \times 100 + 2dk \times 10 + d^{2}.\]

I hope this helps,

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