• 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.

You’re right, I was thinking "if not B then not A" but I said the wrong word. And you’re also right about how I used the converse where I shouldn’t have. Perhaps I should have approached my solution this way.

If the number is a multiple of 3, then it must also be a multiple of 4 (otherwise statement 1 would be true and we would have a contradiction), and must also be a multiple of 6 (otherwise statement 3 would be true and we would have a contradiction). So if the number is a multiple of 3, then it must also be a multiple of 12. And from statement 1, if the number is a multiple of 3, it must between 50 and 59. Combining these two finding we have "If the number is a multiple of 3, then it is both a multiple of 12 and between 50-59". The contrapositive "If the number is not both a multiple of 12 and between 50-59, then the number is not a multiple of 3" is also true. As there is no such number that is both a multiple of 12 and between 50-59, then we can now say that the number must not be a multiple of 3.

We don’t know what group it is in yet, we just know it is not a multiple of 3.

If it is not a multiple of 3 then it is not a multiple of 6.
If it is not a multiple of 6, then it is in the 70-79 range.
If it is in the 70-79 range, then it is not in the 60-69 range.
If it is not in the 60-69 range, then it is not (not a multiple of 4).
So the number is a multiple of 4.

And then again: There are only two multiples of 4 in the 70-79 range: 72 and 76. And since 72 is divisible by 3 and 6, this means the answer is 76.

I think that’s better.