MHB Solving for $X$: Find $X$ When $X=Y^2$

[Albert1]

$X=\overline{abcdef}$ is a 6-digit number,

$Y=\overline{def}$ is a 3-digit number,

if $X=Y^2$ find $X$

 

[kaliprasad]

$X=\overline{abcdef}$ is a 6-digit number,

$Y=\overline{def}$ is a 3-digit number,

if $X=Y^2$ find $X$

Spoiler

let $X = 1000m + Y$
we have $X = Y^2$ or $Y^2-Y= 1000m$ or $Y(Y-1) = 1000m = (125a)(8b)$ where $m = ab$
now $Y,Y-1$ are co-primes so one of them has to be 125a and another 8b and a and b are coprimes or one of them is 1
a has to be odd ( else 125 cannot be coprime to 8b) and < 8 as $Y < 1000$
so we get
$Y=125a, Y-1 = 8b$ trying a = 1,3,5,7 we get $Y= 625, Y-1 = 624 = 8 * 78$
or
$Y=8b, Y-1 = 125a$ we get $Y = 376$
so there are 2 values of X $ 376^2 = 141376$ and $625^2=390625$