• 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.

"Therefore, by the converse of the second statement, we know that the number is not a multiple of 3."

Actually it is the contrapositive of the second statement that leads to this conclusion. When the original premise is "If A then B", the contrapositive, "If not B then not A" is a valid conclusion, and was what was used here.  The converse, "If B then A", is not a valid conclusion.  So the reasoning was valid, but the terminology wrong.

However, the claim "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." does invalidly use the converse in reasoning.  It assumes that statement 1 can be construed as "If the number is between 60 and 69 then it is not a multiple of 4."  This is the converse of what is said, and is therefore not a valid conclusion.  It is the valid remainder of this paragraph that excludes this range: that by the contrapositives of statements 2 and 3 that it must be a multiple of 6 without being a multiple of 3, an impossibility.

Similarly the first statement "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. ": the 3 actually devolves from the 6 in the third statement, and is better stated as "a multiple of 6 and of 4", which still leads to being divisible by 12, but its origin is in the contrapostive of statement 3 rather than the converse of statement 2.