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Q.

Given a three digit number whose digits are three successive terms of a G.P. If we subtract 792 from it, we get a number written by the same digits in reverse order. Now we subtract four from the hundreds digit of the initial number, leaving the other digits unchanged, we get a number whose digits are successive terms of an A.P. Then the product of the digits of given three digit number is ___

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answer is 27.

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Detailed Solution

Let the number be xyz
i) x,y,z are in G.P $\Rightarrow y^{2} = z x$
ii) $x y z = 100 x + 10 y + z$
$x y z - 792 = z y x \Rightarrow x - z = 8$
iii) $x - 4, y, z$ are in AP $\Rightarrow 2 y = x - 4 + 8$
$\Rightarrow {(x - 4 + z)}^{2} = 4 y^{2} = 4 (z x) \Rightarrow {(x + z)}^{2} + 16 - 8 (x + z) = 4 z x \Rightarrow {(x - z)}^{2} - 8 (x + z) + 16 = 0 \Rightarrow x + z = 10 ∴ x = 9, z = 1, y^{2} = 9 \Rightarrow y = 3 ∴ x y z = 931$

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Given a three digit number whose digits are three successive terms of a G.P. If we subtract 792 from it, we get a number written by the same digits in reverse order. Now we subtract four from the hundreds digit of the initial number, leaving the other digits unchanged, we get a number whose digits are successive terms of an A.P. Then the product of the digits of given three digit number is ___