Find X, a 4-digit perfect square

[anemone]

X is a 4-digit perfect square all of whose decimal digits are less than seven. Increasing each digit by three we obtain a perfect square again. Find X.

 

[kaliprasad]

X is a 4-digit perfect square all of whose decimal digits are less than seven. Increasing each digit by three we obtain a perfect square again. Find X.

Spoiler

let the number be $x^2$ and adding 3333 we get $y^2$ and both x and y less than 100. so x + y less than 200

so $y^2-x^2= (y-x)(y+x) = 3333 = 33 * 101$
we need to find product of 2 numbers less than 200 and above is only combination
so y = 67 and x = 34

so $X = 34^2= 1156$
check:
now $y^2 = 67^2 = 4489 = 1156 + 3333$
which is tue

so ans is 1156

 

[anemone]

Spoiler

let the number be $x^2$ and adding 3333 we get $y^2$ and both x and y less than 100. so x + y less than 200

so $y^2-x^2= (y-x)(y+x) = 3333 = 33 * 101$
we need to find product of 2 numbers less than 200 and above is only combination
so y = 67 and x = 34

so $X = 34^2= 1156$
check:
now $y^2 = 67^2 = 4489 = 1156 + 3333$
which is tue

so ans is 1156

Well done, Kali! And thanks for participating!:)