This puzzle amounts
to finding the smallest
exactly sixty-four divisors, counting 1 and the number itself as
The least number is 7,560. The pilgrims might, therefore,
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
not to mention its size, for it certainly would not be possible along
To find how many
different numbers will divide a
N, let N = ap bq cr
..., where a, b, c ... are prime
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
||= 64 divisors.
To find the smallest
number that has a given
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
For example, the smallest number that has seven
divisors and no
more is 64, while 24 has eight divisors, and might equally fulfil the
The stipulation as to
"no more" was not necessary
of my puzzle, for no smaller number has more than sixty-four divisors.