N is a positive base ten integer having at least 2digits but at most 4digits, which is obtained by multiplying the sum of its digits with the product of its digits. It is known that N cannot contain any leading zero.
Determine all possible value(s) of N.
(In reply to
Solution & thoughts by Jer)
I have a start to an analytical solution, too tired to go any further today, perhaps somebody can carry on where I have left off.
Now let s be the sum of the digits of n and p be the product of the digits of n, then we have
n=s*p
we also have that
n mod 9 = s mod 9
so let r = n mod 9 then we have
n=9a+r and s=9b+r
so then we get
9a+r = (9b+r)p
9a+r=9bp+pr
thus we need that
pr mod 9 = r
now pr mod 9 = [r * (p mod 9) ] mod 9
so let p mod 9 = s
then we need
r*s mod 9 = r
now r,s are both on the interval [0,8] and thus we need to find all r,s such that r*s mod 9 = r these are
(r,s): (0,s), (r,1), (3,4), (3,7), (6,4), (6,7)
so what I am thinking can happen is that we can look at each of these possible values for r,s and determine what solutions are possible for n.
Edited on June 21, 2010, 10:58 pm

Posted by Daniel
on 20100621 21:11:59 