First note that Arthur can write down three squares, namely $$16$$, $$25$$ and $$36$$. Also, he can write down four triangular numbers, namely $$15$$, $$21$$, $$36$$ and $$45$$. If he chooses $$16$$ and $$45$$ for the square and triangular number respectively, then the remaining digits are $$2$$ and $$3$$, the prime is $$23$$. If he chooses $$25$$ and $$36$$ then the remaining digits are $$1$$ and $$4$$, the prime is $$41$$. If he chooses $$36$$ for the square number, the remaining difits can be a prime. So the largest prime he write is $$41$$. 
Arthur writes down three two-digit integers. One is square, one is prime and one is triangular.  He uses the digits $$1$$, $$2$$, $$3$$, $$4$$, $$5$$ and $$6$$ exactly once each.  Which largest prime does he write? ． 
Options:
  - $$13 $$ 
  - $$23 $$ 
  - $$31 $$ 
  - $$41 $$ 
  - $$43$$ 