The complex numbers \(z\) and \(w\) have real and imaginary parts given by \(z = a + \mathrm{i}b\) and \(w = c + \mathrm{i}d\). Prove that \(|zw| = |z||w|\).
By considering the complex numbers \(2 + \mathrm{i}\) and \(10 + 11\mathrm{i}\), find positive integers \(h\) and \(k\) such that \(h^2 + k^2 = 5 \times 221\).
Find positive integers \(m\) and \(n\) such that \(m^2 + n^2 = 8045\).
You are given that \(102^2 + 201^2 = 50805\).
Find positive integers \(p\) and \(q\) such that \(p^2 + q^2 = 36 \times 50805\).
Find three distinct pairs of positive integers \(r\) and \(s\) such that \(r^2 + s^2 = 25 \times 1002082\) and \(r < s\).
You are given that \(109 \times 9193 = 1002037\).
Find positive integers \(t\) and \(u\) such that \(t^2 + u^2 = 9193\).