#!/usr/bin/env python3
from functools import lru_cache
from math import isqrt
@lru_cache(None)
def extend(T):
 B=set(T)
 S={x+y for x in B for y in B}
 if 113 in S: return False
 D=S-{x*y for x in B for y in B}
 if not D: return True
 def factors(s):
  return [(u,s//u) for u in range(1,isqrt(s)+1)
          if s%u==0 and s//u<=112 and 113-u not in B
          and 113-s//u not in B and u+s//u!=113]
 s=min(D,key=lambda v:len(factors(v)))
 return any(extend(tuple(sorted(B|{u,v}))) for u,v in factors(s))
A=set(range(1,14))|{16,113}|set(range(15,110,2))
print({a+b for a in A for b in A}<={a*b for a in A for b in A})
print(extend((1,2,3,5,7)))
print(extend.cache_info())