Checking closure of sets of integers under multiplicaton mod(n) [closed]
There are $8$ numbers in the set. Reducing them modulo $(7,13)$:
$1=(1,1)$
$9=(2,9)$
$16=(2,3)$
$22=(1,9)$
$53=(4,1)$
$74=(4,9)$
$79=(2,1)$
$81=(4,3)$
If the set is closed, it should contain the 9 pairs $\{1,2,4\}\times\{1,3,9\}$, so the set is not closed. The missing pair to close the set is $(1,3)$. You can use CRT to convert the pair to the number mod $91$, which is $29$.
Not sure if this is faster than checking $8\choose 2$ pairs though...