The number 496 is the third smallest *perfect* number: the sum of its proper divisors is the number itself:
\[1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248 = 496.\]

But it is also a "perfect" number in the sense that its third digit is the geometric mean of the first two digits: \(\sqrt{4\cdot 9} = 6\).

How many three-digit integers \(\overline{abc}\) are there with the property, that \(c\) is the geometric mean of \(a\) and \(b\)?

**Note**: Numbers starting in zero do not count. Thus, a "three-digit integer" lies between \(100\) and \(999\).

And here is a more challenging variation on the theme.

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