Let A be a point on curve with integral coordinates. Does k.A necessarily have integer coordinates? If so than why and if not than how to find A and k such that k.A has integral coordinates.
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If you look at elliptic curves over $\mathbb C$, and do point addition with points with integer coordinates (as $\mathbb Z \subset \mathbb C$), then the result of the point addition usually will not have integer coordinates.
But in cryptography, we don't use elliptic curves over $\mathbb C$, but over a finite field $\mathbb F$. So the coordinates are not integers, but field elements. Subtracting and dividing field elements by each other (assuming no division by zero) gives you new field elements.
If you represent field elements by integers, then the new coordinates are "integers" again.