## Brain-Teaser 683: Powerless

**From The Sunday Times, 18th August 1974** [link]

I have here two positive single figure numbers, each less than 9. Neither is a factor of the other. I add the larger number to the smaller.

Then, to that total I again add the original larger number, and to the new total I again add the original larger number and may, if I like, continue this process indefinitely, but never shall I obtain a total which is a “power” of any whole number whatsoever.

What are my two numbers?

This puzzle was included in the book **The Sunday Times Book of Brain-Teasers: Book 1** (1980, edited by Victor Bryant and Ronald Postill). The puzzle text above is taken from the book.

[teaser683]

## Jim Randell 8:11 am

on21 January 2021 Permalink |This Python program narrows down the candidate pairs until there is only one left, so if there is a solution this must be it.

It runs in 49ms.

Run:[ @repl.it ]Solution:The two numbers are 6 and 8.To show that this is indeed a solution we need to show that

(6 + 8k)can never be a non trivial exact power.To find a counter example, we are looking for for some positive integers

k, a, n, wheren ≥ 2, such that:Clearly the LHS is a multiple of 2. So the RHS must also be a multiple of 2, so let:

a = 2b.Clearly the RHS is divisible by 2, but the LHS is not divisible by 2.

So

(6 + 8k)can never be an exact power of an integer.Most of the pairs drop out fairly quickly, but (3, 7) hangs in until we get to 2187 = 3 + 312×7 = 3^7.

Leaving (6, 8) as the only remaining candidate.

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## Jim Randell 11:06 am

on21 January 2021 Permalink |Here is a more efficient program that reuses the code I wrote for

Enigma 1777to generate powers in numerical order.Then for each power we remove pairs of numbers that can be used to make that power, until there is only a single pair remaining.

Run:[ @repl.it ]LikeLike

## Frits 8:15 pm

on21 January 2021 Permalink |Python 3.9 introduced multiple arguments version of math.gcd.

Without imports.

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