• if it is not a multiple of 4, then it is between 60 and 69
• if it is a multiple of 3 it is between 50 and 59
• if it is not a multiple of 6 it is between 70 and 79.

This is an interesting problem but is fairly easy to solve.  Since the set of possible solutions is relatively small it is easy yet effective to just list all solutions in the set and cast them out after carefully applying each constraint.  The best way to understand is to follow along on your own sheet of paper.

Constraint 1 initially leaves 61, 62, 63, 65, 66, 67 and 69(none of these are divisible by 4.

Likewise Constraint 2 initially leaves leaves 51, 54, and 57 and Constraint 3 initially leaves 70, 71, 73, 74, 75, 76, 77, and 79.

The new set of possible solutions consists of all these numbers.

Now you can go ahead and apply each constraint across  the groups.  Applying Constraint 1 to groups 2 and 3 eliminates non multpiles of 4 between 50-59 and 70-79, namely 51, 54, 57, 70, 71, 73, 74, 75, 77, and 79.  Now your set of possible solutions is considerable smaller; exactly it consists now of 61, 62, 63, 65, 66, 67, 69, and 76. 

Applying Constraint 2 to what is left of group 2 and 3 eliminates numbers divisible by 3 and now the set of possible solutions is even smaller and consists of 61,62,65,67,and 76.

Finally applying Constraint 3 to what is left of groups 1 and 2 eliminates numbers which are not divisible by 6, taking out 61, 62, 65, and 67.

Now you are only left with 76 which is the answer and satisfies all the necessary conditions.