Either the number is a multiple of 3 or it is not.
If it is then it is between 50 and 59, if it is not, then it is not a multiple of 6 either, and thus it is between 70 and 79. Therefore we know it is not between 60 and 69, and therefore it is a multiple of 4.
If it is between 50 and 59, and is a multiple of 4 it must be 52 or 56.
But neither of these are multiples of 3.
Therefore the number is not between 50 and 59, but between 70 and 79.
Of the numbers between 70 and 79, it can be 72 or 76.
Since 72 is a multiple of 6 however, this leaves 76 as the only possible solution.