If 11|(12i+3j) and 22|j then 11|i.
This is the implication.
Focusing on 22|j. If J is divisible by 22 that means its an even number and is also divisible by 11. Can I go from 22|j to 11|(j/2)?
I dont know what I should do after this point. Should I use Transability of Divisibility or Divisibility of Integer combinations to prove.
$11\mid12i+3j$ means there exists $l$ such that $12i+3j=11l$. If furthermore $j=22k$, then $$12i=11l-3j=11(l-6j),$$ whence $i=12i-11i=11(l-6j-i)$.