123 is a peculiar integer, because 1+2+3=1*2*3. 1412 is also peculiar, since 1+4+1+2=1*4*1*2.

A simple question: are there infinitely many such numbers?

A not so simple question: if so, are there such numbers for ANY number of digits?

lets say you want to find a n digit number such that it meets the said description and that n>=2.

In order to construct such a number start with n-2 1s and let the last two numbers be a and b.  Then we have that their sum is
a+b+n-2
and their product is ab

thus
a+b+n-2=ab
ab-b=a+n-2
b(a-1)=a+n-2
b=(a+n-2)/(a-1)
now if we set a=n then we get
b=(n+n-2)/(n-1)=(2n-2)/(n-1)=2(n-1)/(n-1)=2
b=2

thus we have contrstucted the required number which is made of n-1 1s the number n and the number 2.

Since the only requirement on n is that it is greater than or equal to 2 then we can say that we can construct a "peculiar" number of any length

Posted by Daniel on 2006-08-21 19:33:03