Product and Sum of Digits = Number

Date: 10/24/2001 at 18:58:17
From: Meghan Jordan
Subject: Word problem...

How many two-digit numbers exist such that when the products of their 
digits are added to the sums of their digits, the result is equal to 
the original two-digit number?

Date: 10/25/2001 at 11:05:19
From: Doctor Ian
Subject: Re: Word problem...

Hi Meghan,

A two digit number looks like

  ab

which is just shorthand for 

  10a + b

where a is between 1 and 9, and b is between 0 and 9.  

The product of the digits would be 

  a*b

The sum of the digits would be 

  a+b

So if the number is equal to the product of the digits plus the sum of 
the digits, then

  10a + b = (a*b) + (a+b)

          = a*b + a + b

          = (b+1)*a + b

Now, this is interesting, because it means that b+1 must be equal to 
10. So that certainly narrows things down a bit.  

Can you take it from here? 

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/