The impossibility of exactly simplifying the floor quotient below the square root

Let $p$ be a prime number and let $a$ be an integer satisfying $1<a<p$. Consider the quotient $\left\lfloor \frac{p}{a}\right\rfloor$ for $a\leq\left\lfloor\sqrt{p}\right\rfloor$. Impossibility conjecture. It is impossible to find any exact simplification for

$$\left\lfloor \frac{p}{a}\right\rfloor$$

when $a\leq\left\lfloor\sqrt{p}\right\rfloor$. The claim concerns the distribution of first-order focals in the geometric view of the sieve of Eratosthenes; no resolution or supporting result is supplied in the excerpt.