Problem

Source: Romania National Olympiad 2023

Tags: algebra, number theory



We say that a number n2 has the property (P) if, in its prime factorization, at least one of the factors has an exponent 3. a) Determine the smallest number N with the property that, no matter how we choose N consecutive natural numbers, at least one of them has the property (P). b) Determine the smallest 15 consecutive numbers a1,a2,,a15 that do not have the property (P), such that the sum of the numbers 5a1,5a2,,5a15 is a number with the property (P).