Puzzle of the Week #294 - Special K

K, L and M are all positive whole numbers.

For the certain special values of K that we seek, the same values of L and M that cause (KxL)+(4xM) to be a multiple of 11 also cause (KxM)+(5xL) to be a multiple of 11.

For instance, K ISN’T 2, because some values of L and M that make (2xM)+(5xL) a multiple of 11 (eg L=1,M=3) when you plug those same values of L and M into (2xL)+(4xM) give a number that is NOT a multiple of 11 (in this case 14).

Out of the possible values of K for which the divisibility by 11 of (KxL)+(4xM) and (KxM)+(5xL) are always in agreement, what number is the THIRD LOWEST PRIME?