Checking whether a number is a perfect square or not.

Question

I was given to tell whether $945729$ is perfect square or not. I used the concept that

No number can be a perfect square unless its digital root is $1$, $4$, $7$, or $9$.

Digital root of $945729=9$ but still it is not a perfect square. What am I missing here? Please help !!!

Thanks in advance !!!

Answer 1

What am I missing here?

A number is a perfect square is a sufficient but unnecessary condition of that the digital root is 1, 4, 7, or 9, so it could only tell you that if a number's digital root is NOT 1, 4, 7, or 9, it is NOT a prefect square.

First, the digital root is that the remainder upon division by 9 (except when the digital root is 9),

$(9k+0)^2\ mod\ 9 = 0$ (or to say 9?)

$(9k+1)^2\ mod\ 9 = 1$

$(9k+2)^2\ mod\ 9 = 4$

$(9k+3)^2\ mod\ 9 = 0$

$(9k+4)^2\ mod\ 9 = 7$

$(9k+5)^2\ mod\ 9 = 7$

...

You see, that's how 0,1,4,7,9 comes, but the reverse is not feasible because it's mod here, like 7 is not a perfect square itself.

Then what is the easiest way to find whether the number is a perfect square or not.

Finding square roots by hand may be helped.

Perfect square trinomial could helped if number is too big,

You could check it out at Calculate A Square Root, the 2nd method on it, this might not be easy but should be useful for competitive examinations in my opinion.