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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