Answer :
This puzzle amounts
to finding the smallest
possible number
that has
exactly sixtyfour 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 sixtyfour 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} 
× 
3^{3} 
× 
5 
× 
7 
Powers = 
3 

3 

1 

1 

Therefore 
4 
× 
4 
× 
2 
× 
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 sixtyfour divisors.
