Question

The digits of a three-digit number form a G.P. If $400$ is subtracted from it, then we get another three-digit number whose digits form an arithmetic series. What is the sum of these two numbers?

• A. $1356$
• B. $1648$
• C. $1462$
• D. $1000$

Answer 1

The correct option is C $1462$

Let the digits of a three-digit number be $x,y$ and $z$ and the number be $100z+10y+x$, where $x,y,z$ are in G.P.
$\therefore y^2=xz$
$\Rightarrow xz$ must be a square number

If $xz=4$, then $y=2\Rightarrow 421$ is the number, but $421-400=21$ which is not a three digit number.
$\therefore xz$ $=$ $9$, i.e., $x$ $=$ $1$, $z$ $=$ $9$
$\therefore y=3$, so that $x,y,z$ are in A.P.
$\therefore$ The number is $931$.
The other number will be $531$, so that $1,3$ and $5$ are in A. P.

Their sum $=$ $931+531$ $=$ $1462$.