13 and 117

Algebra Level 2

Let $a, b, c$ ... be rational numbers greater than 1.

Can we find such a set that added together equals 13 and multiplied together equals 117?

(Example: If we wanted a set that added to 9 and multiplied to 25, we could pick 2, 2, 2.5 and 2.5)

Yes Cannot be determined No

1 solution

Denton Young
Jan 21, 2020

The set (2.4, 2.5, 2.5, 2.6, 3) meets the requirements.

Nice question. I first checked to see what the maximum product of $n$ such rational numbers which added to 13 could possibly be. This is achieved when all $n$ numbers are the same value $a = \dfrac{13}{n}, n \lt 13$ . For a non-integral value of $n$ the maximum of $a^{n} = \left(\dfrac{13}{n}\right)^{n}$ is $e^{13/e} \approx 119.4 \gt 117$ , which prompted me to go ahead and guess 'Yes', but I wasn't able to prove to myself that this could be achieved with rational numbers only, as your example does. (Note that for $a = 13/5 = 2.6$ , roughly in line with your example, we get $a^{5} \approx 118.8$ , which suggests that any solution set would have to involve 5 rational numbers) I'll have to give this more thought. Anyway, a follow-up question would be the same but with 120 (or even 119) replacing 117, the answer to which would definitely be 'No'.

Brian Charlesworth - 1 year, 4 months ago

I had the same approach in terms of estimation. It's interesting that there have to be exactly $5$ numbers - both $\left(\frac{13}{4}\right)^4$ and $\left(\frac{13}{6}\right)^6$ are too small. Are there any other sets of rationals that work? Are there infinitely many? Are there examples of similar questions where this estimation technique doesn't work? (ie we're looking for a set of rationals with sum $S$ , product $P<e^{S/e}$ , but none exists).

Chris Lewis - 1 year, 4 months ago

To answer your second question, Brian gave an example. 13 and 119: e^(13/e) is > 119, but 2.6^5 < 119.

Denton Young - 1 year, 4 months ago

Partly answering my own question: the set $2. \dot{2},2.6,2.7,2.7,2.\dot{7}$ also works.

Chris Lewis - 1 year, 4 months ago

13 and 117

1 solution

0 pending reports