Question:

The prime factorization of 2001 is determined as follows: $$2001 = 3 \times 667 = 3 \times 23 \times 29$$ Given that \(a, b, c\) are prime factors of 2001 such that \(a < b < c\), we have: $$a = 3, \quad b = 23, \quad c = 29$$ The direction ratios of a normal to the plane are given as \(b, c, a\). Therefore, the direction ratios are \(23, 29, 3\). The equation of a plane passing through a point \((x_0, y_0, z_0)\) with normal direction ratios \((A, B, C)\) is given by \(A(x - x_0) + B(y - y_0) + C(z - z_0) = 0\). Given the plane passes through the point \((1, 1, 1)\) and has normal direction ratios \((23, 29, 3)\), its equation is: $$23(x - 1) + 29(y - 1) + 3(z - 1) = 0$$ Expanding and simplifying the equation: $$23x - 23 + 29y - 29 + 3z - 3 = 0$$ $$23x + 29y + 3z = 23 + 29 + 3$$ $$23x + 29y + 3z = 55$$