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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


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 


The sum of the digits would be 


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
Associated Topics:
High School Number Theory
High School Puzzles
Middle School Puzzles

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