Evaluating Mathematical Statements

Statement I: $1000027 = 7 \times 103 \times 1387 = 7 \times 103 \times 19 \times 73$, so 19 is also a prime factor (False). Statement II: Raising to power 42 gives $(6!)^7 > (7!)^6 \implies (6!)^7 > 7^6 (6!)^6 \implies 6! > 7^6 = 117649$, which is False ($720 < 117649$). Statement III: Average speed $V = \frac{2 v_1 v_2}{v_1 + v_2} = \frac{2 x y}{x + y}$. For $V = 2x$, $\frac{2 x y}{x + y} = 2x \implies y = x + y \implies 0 = x$, impossible. Thus Statement III is True.