MathDB

We say that a number

n \ge 2

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

a_1, a_2, \ldots, a_{15}

that do not have the property

(P),

such that the sum of the numbers

5 a_1, 5 a_2, \ldots, 5 a_{15}

is a number with the property

(P).

Problem Statement