Problem Solving Set 14

17 July 2012

1. Denote by V the real vector space of all real polynomials in one variable, and let P : V → ℝ be a linear map. Suppose that for all f, g ∈ V with P (fg) = 0 we have P (f) = 0 or P (g) = 0. Prove that there exist real numbers x_{0}, c such that P (f) = cf(x_{0}) for all f ∈ V.

2. The smallest number with four factors is 6 : 1, 2, 3, 6. The smallest number with six factors is 12 : 1, 2, 3, 4, 6, 12. What is the smallest number with 100 factors?