# How to find the numbers that the product of it's digits is equal to ten times the sum of them.

I formulated this so that the number be in range $[111,999]$, it was narrowed so that $a,b,c$ is not equal to zero.
$$a\cdot b\cdot c = (a+b+c)\cdot 10$$ With this we can see that $\frac{a\cdot b\cdot c}{10} = a+b+c$ and so the product of $a\cdot b\cdot c$ should be divisible by $10$. Right now I know that one number must be $5$ and the other one must be even.

$$\frac{5bc}{10}=(5+b+c)$$

To narrow it more the maximum product can be $$360 (5\cdot 8\cdot 9)$$

Now I have $36$ numbers that meets my conditions

Is it possible to narrow the search even more?

• $bc$ has to be more than $10$ – Empy2 Apr 25 '15 at 16:56
• $5+b+c\leq22$ so $bc\leq44$ – Empy2 Apr 25 '15 at 16:58
• @Michael Why bc has to be more than 10? – Juanvulcano Apr 25 '15 at 17:02
• because $5bc/10>5$ – Empy2 Apr 25 '15 at 17:05

When you deduced that $a=5$ then: $$bc=2(b+c+5) \iff (b-2)(c-2)=14$$