Squaring by conjugation

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

Find abc.\overline{abc}.

Details and assumptions

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

The number 12=01212=012 is a 2-digit number, not a 3-digit number.


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