Problem Statement
Let X be the group generated by and , such that
X=\langle u, v \mid u^{46} = 1, v^{40,488,236} = 1, vu = uv^{27,636,633} \rangle $$. What is the order of the group X?Ground Truth
184
Execution Trace
Reasoning Process
[THOUGHT] We have a metacyclic presentation with relations u^46=1, v^N=1, and vu = u v^m, which implies u^{-1} v u = v^m. Since u^46=1, conjugation by u^46 is trivial, so v = u^{-46} v u^{46} = v^{m^46}. Hence v^{m^46-1}=1. Together with v^N=1, the order of v divides gcd(N, m^46-1). Let d = gcd(N, m^46-1) = gcd(N, (m^46 mod N)-1). Then the group has order 46*d. [/THOUGHT] [PYTHON] import math
N = 40488236 m = 27636633
r = pow(m, 46, N) # m^46 mod N d = math.gcd(N, r - 1) # gcd(N, m^46 - 1)
print("m^46 mod N =", r) print("d =", d) print("group order =", 46*d) [/PYTHON]
Tool Output
[PYTHON OUTPUT]
m^46 mod N = 22177545
d = 4
group order = 184
[/PYTHON OUTPUT]
Reasoning Process
[THOUGHT] From the computation, (d=\gcd(N,m^{46}-1)=4), so (|X|=46\cdot 4=184). [ \boxed{184} ]
Final Model Output
184
Comparative Analysis (P0) (Average across runs)
Avg Time Taken
Solar-Open-100B
927.7
K-EXAONE-236B-A23B
261.4
Kanana-2-30B-Thinking
94.3
GLM-4.5-Air
89.6
Gemini-3-Pro-Preview
83.6
Kanana-2-30B-Thinking-2601
73.2
Solar-Pro 2
63.9
GPT-5.2 (high)
44.4
naver-hyperclovax/HCX-007
31.0
model_d_r1
16.8
axk1
6.7
EXAONE-4.0-32B
3.8
Avg Token Usage
Solar-Open-100B
49491.3
K-EXAONE-236B-A23B
19786.0
Kanana-2-30B-Thinking-2601
16432.3
Kanana-2-30B-Thinking
13797.7
Solar-Pro 2
13080.0
Gemini-3-Pro-Preview
11033.7
GLM-4.5-Air
9584.0
GPT-5.2 (high)
4265.7
naver-hyperclovax/HCX-007
4078.0
model_d_r1
3423.7
EXAONE-4.0-32B
3113.7
axk1
2637.3