I'm thinking of a number.

- 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.

What is the number?

If possible, let us assume that 60, 64 and 68 which are multiples of 4 correspond to valid solutions.Then, 64 and 68 are not multiples of 6 and yet, they do not lie in the range 70 to 79. This is a contradiction.Also, 60 is a multiple 0f 3, not lying in the range 50 to 59. This is another contravention.

Hence, all the numbers divisible by 4 cannot lie in the range 60 to 69 and vice versa. Now, 51, 54 and 57 are the numbers divisible by 3 in the range 50 to 59, but none of these three numbers are divisible by 4. This is a contradiction.Hence, all NON MULTIPLES of 3 must lie in the range 50 to 59 and vice versa.

Now, in the range 70 to 79:

(a) The only number which is not a multiple of 6 but correspond to a multiple of 3 is 75. This is not a multiple of 4 and yet do not lie in the range 60-69. Contradiction.

(b) Numbers which are multiples of 4 correspond to 72 and 76. But, 72 is a muliple of 6. This is a contradiction. Consequently, the required number must be 76.