#!/usr/bin/env python3
from functools import lru_cache
from math import isqrt
N=112
@lru_cache(None)
def extend(T):
 B=set(T)
 S={a+b for a in B for b in B}
 if 113 in S:return False
 D=S-{a*b for a in B for b in B}
 if not D:return True
 def factors(s):
  return [(d,s//d) for d in range(1,isqrt(s)+1)
          if s%d==0 and s//d<=N and 113-d not in B
          and 113-s//d not in B and d+s//d!=113]
 s=min(D,key=lambda v:len(factors(v)))
 return any(extend(tuple(sorted(B|{u,v}))) for u,v in factors(s))
print(extend((1,2,3,5,7)),extend.cache_info())