There exists a solution to $S_10(x) = x/n$ for every integer $n$, assuming $S_10(x)$ is the sum of all base 10 digits in $x$

Claude Opus 4.7

Vote: No

We need S_10(x) = x/n for every integer n, meaning for each n there exists a positive integer x with x = n·S_10(x). For n=1: need x = S_10(x), works for x=1 through 9. However, consider large n. For any x, S_10(x) ≤ 9·(number of digits of x) ≈ 9·log10(x)+9. So x/S_10(x) grows roughly like x/(9 log x), which can be made arbitrarily large, so large n seem achievable. But we must check every integer n, including n=0 and negative n. For n=0: S_10(x) = x/0 is undefined, so no solution exists. Also for negative n, S_10(x) is non-negative for positive x, and x/n would be negative — requiring S_10(x)=0, meaning x=0, but then n·0=0=x works trivially with x=0. Still, n=0 is the killer: division by zero. Even setting that aside, the statement says 'every integer n', and n=0 makes the equation undefined. So the answer is No.

GPT-5.5

Vote: No

The statement says “for every integer n.” But for n = 0, the expression x/n is undefined, so the equation S_10(x) = x/n cannot even be formed, let alone solved. Therefore the universal claim is false as stated, so the correct choice is "B: No".

Grok 4.20

Vote: No

The given statement claims that there exists a solution for every integer n. However, integers include n = 0. For n = 0, the right-hand side x/0 is mathematically undefined, so no value of x can satisfy the equation S_10(x) = x/0. This alone makes the statement false, regardless of how S_10(x) is defined or whether solutions exist for nonzero n. Even if x = 0 is considered, 0/0 remains indeterminate and cannot equal S_10(0) = 0.

Final Summary

The AI Roundtable encountered a classic mathematical 'gotcha' and responded with unanimous rejection. Claude Opus 4.7, GPT-5.5, and Grok 4.20 all immediately pinpointed that the inclusion of $n=0$ in the set of 'every integer' makes the expression $x/n$ undefined, rendering the universal claim false by default. Gemini 3.1 Pro was unable to join the consensus, failing to provide a response within the time limit.

No agreement after 1 rounds of discussion. Final vote: No: 3

Strongest Arguments

• No: The statement asserts a solution exists for every integer $n$; however, for $n=0$, the term $x/n$ is mathematically undefined, meaning no solution can exist for that specific case.