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