An algebraic riddle: The king's chest full of bags of gold coins

Suppose there were $182$ coins. That could be $14×13$ or $26×7$, among other things. So we could not tell how many bags or how many coins per bag there are.

Suppose there were $187$ coins. That can only be $11×17$, the product of two primes. But which is $11$, the number of bags or the number of coins per bag?

To avoid these ambiguities, you need a number of coins that is the product of two identical primes, meaning it's the square of this common prime. There is only one such number between $150$ and $200$.

Apparently the king is not triskaidekaphobic ... .


Let's note $b$ the number of bags in the chest.

Let's note $c$ the number of coins in each bags.

Then we can easily deduce that the total number of coins $t$ in the chest is :

$$t = b*c$$

Furthermore we have $b>1,c>1$ as a condition.

Now the problem here is that if the King gives $t$ then we end up with one equation with two variables, meaning there can be multiple solutions :

For example, let say the King says "There is $180$ coins in the chest" which is $t=180$, now we end up with the following equation :

$$180 = b*c$$

For which there is multiple $(b,c)$ solutions :

$$(2,90),(3,60),(4,45), (5,36),(6,30),(9,20),(10,18),(12,15),(15,12),(18,10),(20,9,(30,6),(36,5),(45,4),(60,3),(90,2)$$

Here I chose $180$ to show there could be a lot of solutions.

There could also be no solution if the King says a prime number between $150$ and $200$, which are :

$$151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199$$

Because the number of bags and coins is an integer and any of those numbers cannot be the product of two integers greater than $1$, there is no solution for those numbers.

EDIT :

If I'd tell you the total of coins in the chest, you'd be able to tell me how >many bags the chest holds and how many coins each bag holds.

This means that the total number of coins gives a single $(b,c)$ solution. Which means :

  • $b=c$, or else $(b,c)$ and $(c,b)$ are two different solutions.

  • $b$ is a prime number, or else we could have more solutions.

Now with those clues and $t\in [150,200]$ we can deduce that :

$$t=169\text{ and } b=c=13$$

Tags:

Inequality

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