What’s My Number

If I am not a multiple of $4$, then I am between $60$ and $69$.
If I am a multiple of $3$, I am between $50$ and $59$.
If I am not a multiple of $6$, I am between $70$ and $79$.
What number am I?
Source: CMC Communicator Volume 29, No. 3 March 2005

SOLUTION
Not a multiple of $4$ between $60$ and $69$:
$61, 62, 63, 65, 66, 67 \qquad\left (1\right )$
Multiples of $3$ between $50$ and $59$:
$54, 57 \qquad\qquad\qquad\qquad\left (2\right )$
Not a multiple of $6$ between $70$ and $79$:
$71, 73, 74, 75, 76, 77 \qquad\left (3\right )$
We now examine each candidate number and look for those that pass all three conditions. When a number passes/fails a condition we mark it by a P/F in the following table

Of the candidate numbers only 76 passes all three conditions.