(a) To find \(w\), use the conjugate of \(1 + 3i\):

\(w = \frac{2 + 4i}{1 + 3i} \times \frac{1 - 3i}{1 - 3i} = \frac{7 - i}{5}\)

Square \(w\) to find \(w^2\):

\(w^2 = \left(\frac{7 - i}{5}\right)^2 = \frac{48 - 14i}{25}\)

Find the modulus of \(w^2\):

\(|w^2| = \sqrt{\left(\frac{48}{25}\right)^2 + \left(\frac{-14}{25}\right)^2} = 2\)

Find the argument of \(w^2\):

\(\arg(w^2) = \arctan\left(\frac{-14}{48}\right) = -0.284 \text{ radians or } -16.3^\circ\)

(b) The locus \(|z| = 5\) is a circle centered at the origin with radius 5. The locus \(|z - 5| = |z|\) is a perpendicular bisector of the line segment from the origin to the point (5,0), which is a vertical line through \(x = 2.5\).

The points common to both loci are where the circle intersects the line:

\(z = 5e^{\pm \frac{1}{3} \pi i}\)