Question 17
We say that a natural number $n \ge 2$ is *nice*, if in its prime factor decomposition, one of the primes has exponent 3.
a) Determine the smallest natural number $N$, such that no matter how we choose $N$ consecutive natural numbers, at least one of them is nice.
b) Find the smallest 15 consecutive natural numbers $a_1, a_2, \dots, a_{15}$ that are not nice, but for which the sum $5a_1 + 5a_2 + \dots + 5a_{15}$ is nice.
a) Determine the smallest natural number $N$, such that no matter how we choose $N$ consecutive natural numbers, at least one of them is nice.
b) Find the smallest 15 consecutive natural numbers $a_1, a_2, \dots, a_{15}$ that are not nice, but for which the sum $5a_1 + 5a_2 + \dots + 5a_{15}$ is nice.