This is my first time posting so do correct me if I am doing anything wrong.

Please help me with this math problem from the British Maths Olympiad (1994 British Maths Olympiad1 Q1 Number Theory).

Starting with any three digit number $n$ (such as $n = 625$) we obtain a new number $f(n)$ which is equal to the sum of the three digits of $n$, their three products in pairs, and the product of all three digits. Find all three digit numbers such that $\frac{n}{f(n)}=1$.

The only solution I found is $199$, can someone verify it please?

  • $\begingroup$ in the definition of $f$, are you concatenating the three results? Can you give an example of a pair $(n,f(n))$? $\endgroup$ – mathworker21 Nov 19 '18 at 5:44
  • $\begingroup$ So $f(199) = 19, 9918, 81$? $\endgroup$ – steven gregory Nov 19 '18 at 5:45
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    $\begingroup$ @mathworker21`@stevengregory For example if $n=625$, $f(n)=6+2+5+6*2+6*5+2*5+6*2*5$ $\endgroup$ – 3684 Nov 19 '18 at 5:51
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    $\begingroup$ to do this problem, write $n = 100a+10b+c$ and just write everything out and solve the equation you get $\endgroup$ – mathworker21 Nov 19 '18 at 5:56
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    $\begingroup$ @mathworker21, I have tried that but I didn't get too far, can you try it if you have time? I also thought about factorising a+b+c+ab+ac+bc+abc as (a+1)(b+1)(c+1)-1 but still wasn't able to get far. For number theory Olympiad problems are there systematic methods or does it require a different method every time. $\endgroup$ – 3684 Nov 19 '18 at 6:00

Let $n=100a+10b+c,$ where $a> 0$ and $b,c\geq 0$. We are trying to solve $$100a+10b+c=a+b+c+ab+ac+bc+abc \\ \implies 99a+9b=abc+ab+ac+bc \\ \implies a(99-b-c-bc)=b(c-9) \\$$$c-9\leq 0$, but $b+c+bc\leq 99$. So the above equation holds iff $b=c=9$, which means $a$ can take any value.

  • $\begingroup$ May I ask how you got to the solution so quick, do you just see the solution? $\endgroup$ – 3684 Nov 19 '18 at 6:13
  • $\begingroup$ @3684 this problem is fairly 'routine'. Plus, I might've seen this before/its associated solution, but I wouldn't remember if I did $\endgroup$ – user574848 Nov 19 '18 at 6:15
  • $\begingroup$ How many main 'routine' methods to solve number theory problems would you say there are? At higher levels do you think each question requires some different insight. $\endgroup$ – 3684 Nov 19 '18 at 6:19

Here's part $(b)$ because I'm assuming you don't need help with part $(a)$:

We want to compute all possible integers $n$ such that $\frac{n}{f(n)} = 1$. Since we know that, by assumption, $n$ is a three-digit number, we can write

$$n = 100a + 10b + c,$$

where $a, b, c$ are integers. If this is the case, in terms of our newly defined variables $a$, $b$, and $c$, we can express $f(n)$ as follows:

$$f(n) = abc + ab + bc + ac + a + b + c.$$

Now, in order to have $\frac{n}{f(n)} = 1,$ we must have $n = f(n)$. This happens when

$$99a + 9b = abc + ab + bc + ac$$

$$\Longleftrightarrow (9-c)b = a(bc + b + c - 99) $$

Also, we must have $b, c \leq 9,$ which implies $bc + b + c - 99 \leq 0$. However, since $a \neq 0$ (if $a = 0$, we would be able to form a two or one-digit number instead of a three-digit one!), we conclude $b = c = 9$. Therefore, our solution set is given by

$$\boxed{\{199, 299, 399, 499, 599, 699, 799, 899, 999\}}$$

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    $\begingroup$ Thank you for your answer, sorry I could only accept one answer. May I ask how you see the solution so quick? $\endgroup$ – 3684 Nov 19 '18 at 6:15

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