This puzzle amounts to finding the smallest possible number that has
exactly sixty-four divisors, counting 1 and the number itself as
divisors. The least number is 7,560. The pilgrims might, therefore, have
ridden in single file, two and two, three and three, four and four, and
so on, in exactly sixty-four different ways, the last manner being in a
single row of 7,560.
The Merchant was careful to say that they were going over a common, and
not to mention its size, for it certainly would not be possible along an
ordinary road!
To find how many different numbers will divide a given number, N, let N =
_a_^p _b_^q _c_^r ..., where _a_, _b_, _c_ ... are prime numbers. Then
the number of divisors will be (_p_ + 1) (_q_ + 1) (_r_ + 1) ..., which
includes as divisors 1 and N itself. Thus in the case of my puzzle--
7,560 = 2^3 x 3^3 x 5 x 7
Powers = 3 3 1 1
Therefore 4 x 4 x 2 x 2 = 64 divisors.
To find the smallest number that has a given number of divisors we must
proceed by trial. But it is important sometimes to note whether or not
the condition is that there shall be a given number of divisors _and no
more_. For example, the smallest number that has seven divisors and no
more is 64, while 24 has eight divisors, and might equally fulfil the
conditions. The stipulation as to "no more" was not necessary in the case
of my puzzle, for no smaller number has more than sixty-four divisors.