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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

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

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