The city wants to have street lights along a street that is $64$ yards long. If the street lights are going to be apart at equal distances that are whole numbers of yards, how may different ways can the city do it?
Can someone please shed some lights on this question (no pun intended)? I divided 64 yards by $1$ to $64$ and the whole numbers from the division are $1$ yard, $2$ yards, $4$ yards, $8$ yards, $16$ yards, $32$ yards and $64$ yards. Is there a much quicker way to do this for any number $n$ of yards, i.e. a formula perhaps?
Thank you very much for your help.
The question is asking how many factors does $64$ have
$$64=2^6$$
Hence the factors are $2^0, 2^1, \ldots, 2^6$. There are $7$ factors.
In general I will consider the prime factorization of the number $n$.
$$n = \prod_{i=1}^q p_i^{m_i}$$
and the number of factors would be $\prod_{i=1}^q (m_i + 1)$