# Squaring by conjugation

Suppose $$\overline{abc}$$ and $$\overline{def}$$ are three-digit numbers such that $(\overline{abc}+\overline{def})^2=\overline{abcdef}.$

Find $$\overline{abc}.$$

Details and assumptions

$$\overline{abc}$$ means $$100a + 10b + 1c$$, as opposed to $$a \times b \times c$$. As an explicit example, for $$a=2, b=3, c=4$$, $$\overline{abc} = 234$$ and not $$2 \times 3 \times 4 = 24$$.

The number $$12=012$$ is a 2-digit number, not a 3-digit number.

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