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$U$, $V$, $C$ , $D$ , $E$, $F$, $G$, $H$, are distinct digits, varying from 1 to 9.

$UVCD$, $EFGHD$, $EHHEV$ are concatenated numbers.

Please deduce these with concise reasoning from the given relations:

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First, solve the first equation

Sqrt(D) is an integer, so must be 1-3. 27^2 is only 3 digits, so Sqrt(D) is 3, and D is 9. X^3 % 10 is 9, so X is 9, so U+V+C % 10 is 9. U+V+C is 19.
6859 = (6+8+5)^sqrt(9)

Then, the last equation

EHHE8 = E^2 * H^H! * 8
H<=3, because otherwise H^H! is way too big H>2 because otherwise the total can't reach 5 digits, so H=3, that leaves E=2.
23328 = 2 * 3^3! * 2 * 8

And now to attack the middle equation

2FG39 = 2^F + G! + 3^9
F*1000 + G*100 + 20039 = 2^F + G! + 3^9
F*1000 + G*100 + 356 = 2^F + G!
G=7, (only 4 digit factorial) so F = 4
24739 = 2^4 + 7! + 3^9

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  • 1
    $\begingroup$ Excellent logic and deduction!!! $\endgroup$ – Uvc Jun 24 at 15:50

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