# The product of digits equal to the sum of digits

How to find the number(or numbers ) that has $4$ digits, the product of these digits equal to the sum of these digits ?

• Not sure what does this have to do with the number itself? You are just looking for 4 digits that summed or multiplied give the same result? – gt6989b Nov 2 '12 at 13:21
• @gt6989b yes but may be there is more than one such number . – htm Nov 2 '12 at 13:25
• I'd think if there is one, you could get other ones by rotating them? I.e. if 1234 is ok, then 4321 must be ok as well, no? Or am I misunderstanding something? – gt6989b Nov 2 '12 at 13:26
• @gt6989b yes you are right . – htm Nov 2 '12 at 13:32
• I wonder why everybody is excluding zero as a digit up front, 0000 seems like an obvious first shot to me? – Frerich Raabe Nov 19 '12 at 9:22

First of all, let's observe that all of the digits of such a number cannot be the same. You can just manually check that numbers $1111$, $2222$ and so on don't suit us. It is also clear that all of the digits should be non-zero.

Now suppose that we have such a number. Let $a,\,b,\,c,\,d$ be its digits written in non-ascending order: $a \geqslant b \geqslant c \geqslant d$. Then we have $$abcd = a + b + c + d.$$

From this we have an inequality: $$a\cdot bcd < 4a.$$ This inequality is strict, because at least one of $b, c, d$ is strictly smaller than a. So we have: $$bcd < 4,$$ which is the same as saying $$bcd \leqslant 3.$$ This only leaves us with 3 possible combinations for $(b, c, d)$: $(1, 1, 1)$, $(2, 1, 1)$ and $(3, 1, 1)$.

If $b=c=d=1$, then $a\cdot 1 \cdot 1 \cdot 1 = a + 1 + 1 + 1$, which can't be true.

If $b=2$ and $c=d=1$, then $a \cdot 2 \cdot 1 \cdot 1 = a + 2 + 1 + 1$, which means that $a=4$. This gives us one possible solution: $a=4, b=2, c=d=1$.

If $b=3$ and $c=d=1$, then $a \cdot 3 \cdot 1 \cdot 1 = a + 3 + 1 + 1$, which is impossible.

So, the only solution is $a=4$, $b=2$, $c=d=1$. There are $12$ numbers with such digits.

• Why do you exclude zero up front, 0000 would work as well, no? – Frerich Raabe Nov 19 '12 at 9:24
• @FrerichRaabe I think that technically $0000$ is not a 4-digit number. It is my understanding that 4-digit numbers all lie between $1000$ and $9999$. $0050$, for instance, is a 2-digit number, because its standard form is $50$. From this point of view, $0000$ is a 1-digit number, because the standard notation for it is $0$. – Dan Shved Nov 19 '12 at 11:22

You can narrow your search rapidly:

1. no digits $0$;
2. at least one digit $1$ (otherwise the product exceeds the sum easily);
3. at least two digits greater than $1$ (otherwise the sum now exceeds the product);
4. exactly two digits greater than $1$ (the product of three such digits would exceed their sum by at least $2$).

So we're looking for pairs of digits in $\{2,3,\ldots,9\}$ whose product exceeds their sum by exactly $2$ (the number of digits $1$ we need to throw in). If one of them is $2$, the other must be $4$. If the smallest of the pair is at least $3$, then their product exceeds their sum by at least $3$, so this cannot happen.

So all in all there is essentially one solution, but since you asked for numbers , the $12$ permutations of the digits of $1124$ give you all solutions.

i observed a pattern in these numbers. 22 123 1124 11125 111126 1111127 and derived a formula for this.

if the last two digits are assumed to be a and b and for an n digit number there will be n-2 1's and a and b are to be found out using the below formula.

a=(b+n-2)/(b-1)

where a and b are from 2 to 9 which need to be evaluated manually for b=2 to 9

• Does this give a number as the question asks? – vonbrand Jan 29 '13 at 17:32
• Yes.. if the number of digits is 4 a=(b+2)/(b-1) and number of 1's in the number is 2.so the number would be 11ab.Now we have to solve for a and b using a=(b+2)/(b-1).which gives integer values for b=2 or b=4 and a=4 when b is 2 and a=2 when b=4.so the number would be 1124 and all permutations of it like 1124,1142,1214,1412 and so on. – Syam Kumar Feb 1 '13 at 7:56

Must be various combinations of $1,1,2,4$. I don't think there are any other combinations, I looked at all small numbers...

• Can we find the number by using algebraic method ? – htm Nov 2 '12 at 13:41
• Yes we can find those numbers using algebraic method – user90533 Mar 24 '16 at 3:32