{{begin-eqn}} {{eqn | l = 6       | r = 3 + 3       | c = $1$ way }} {{end-eqn}} {{begin-eqn}} {{eqn | l = 8       | r = 5 + 3       | c = $1$ way }} {{end-eqn}} {{begin-eqn}} {{eqn | l = 10       | r = 7 + 3       | c = $2$ ways }} {{eqn | r = 5 + 5       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = 12       | r = 7 + 5       | c = $1$ way }} {{end-eqn}} {{begin-eqn}} {{eqn | l = 14       | r = 11 + 3       | c = $2$ ways }} {{eqn | r = 7 + 7       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = 16       | r = 13 + 3       | c = $2$ ways }} {{eqn | r = 11 + 5       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = 18       | r = 13 + 5       | c = $2$ ways }} {{eqn | r = 11 + 7       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = 20       | r = 17 + 3       | c = $2$ ways }} {{eqn | r = 13 + 7       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = 22       | r = 19 + 3       | c = $3$ ways }} {{eqn | r = 17 + 5       | c =  }} {{eqn | r = 11 + 11       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = 24       | r = 19 + 5       | c = $2$ ways }} {{eqn | r = 13 + 11       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = 26       | r = 23 + 3       | c = $3$ ways }} {{eqn | r = 19 + 7       | c =  }} {{eqn | r = 13 + 13       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = 28       | r = 23 + 5       | c = $2$ ways }} {{eqn | r = 17 + 11       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = 30       | r = 23 + 7       | c = $3$ ways }} {{eqn | r = 19 + 11       | c =  }} {{eqn | r = 17 + 13       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = 32       | r = 29 + 3       | c = $2$ ways }} {{eqn | r = 19 + 13       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = 34       | r = 31 + 3       | c = $4$ ways }} {{eqn | r = 29 + 5       | c =  }} {{eqn | r = 23 + 11       | c =  }} {{eqn | r = 17 + 17       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = 36       | r = 31 + 5       | c = $4$ ways }} {{eqn | r = 29 + 7       | c =  }} {{eqn | r = 23 + 13       | c =  }} {{eqn | r = 19 + 17       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = 38       | r = 31 + 7       | c = $2$ ways }} {{eqn | r = 19 + 19       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = 40       | r = 37 + 3       | c = $3$ ways }} {{eqn | r = 29 + 11       | c =  }} {{eqn | r = 23 + 17       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = 42       | r = 37 + 5       | c = $4$ ways }} {{eqn | r = 31 + 11       | c =  }} {{eqn | r = 29 + 13       | c =  }} {{eqn | r = 23 + 19       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = 44       | r = 41 + 3       | c = $3$ ways }} {{eqn | r = 37 + 7       | c =  }} {{eqn | r = 31 + 13       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = 46       | r = 43 + 3       | c = $4$ ways }} {{eqn | r = 41 + 5       | c =  }} {{eqn | r = 29 + 17       | c =  }} {{eqn | r = 23 + 23       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = 48       | r = 43 + 5       | c = $5$ ways }} {{eqn | r = 41 + 7       | c =  }} {{eqn | r = 37 + 11       | c =  }} {{eqn | r = 31 + 17       | c =  }} {{eqn | r = 29 + 19       | c =  }} {{end-eqn}} From Smallest Positive Integer which is Sum of 2 Odd Primes in 6 Ways, the smallest positive integer which is the sum of $2$ odd primes in $6$ different ways is $60$. {{qed}}