• 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.

The answer is 76. Here’s how I got there.

In order for the number to be in the 50-59 range, the number must both be a multiple of 3 and a multiple of 4. (If it weren’t a multiple of 4, it would have to be in the 60-69 range.) So if the number is in the 50-59 range, it must be a multiple of 12. There are no multiples of 12 between 50 and 59, so the number is not in this range. Therefore, by the converse of the second statement, we know that the number is not a multiple of 3.

Next, let’s look at the 60-69 range. In order for the number to be in this range, it must NOT be a multiple of 4, but it MUST be a multiple of 6. (If it weren’t a multiple of 6, it would have to be in the 70-79 range.) But if the number is divisible by 6 then it is also divisible by 3. And we already said the number couldn’t be divisible by 3.

Therefore the number must be in the 70-79 range, and the number is not a multiple of 6 (or 3 for that matter) but IS a multiple of 4. There are only two multiples of 4 in this range: 72 and 76. And since 72 is divisible by 6, this means the answer is 76.

The number can’t be outside of the 50-79 range. If it were, it would have to be divisible by 6 (otherwise it would have to be in the 70-79 range). But if it is divisible by 6 then it is divisible by 3 and would have to be in the 50-59 range.

So the answer is 76.