SOLUTION: Find a six-digit multiple of $64$ that consists only of the digits $2$ and $4$.

Notice that 64 = .


(1)  Since the number N is a multiple of 64, it is a multiple of 4, too.

     Hence, its two right-most digits constitute the number, multiple of 4.

     The only possible two-digit numbers consisting of digits 2 and 4 and multiple of 4 are 24 and 44.

     So, the two right-most digits of the number N are  "24"  or  "44".



(2)  Since the number N is a multiple of 64, it is a multiple of 8, too.

     Hence, its three right-most digits constitute the number, multiple of 8 
            (according to the rule of divisibility by 8).

     In addition, we know that its two right-most digits are "24"  or  "44".

     The only  possible such three-digit numbers consisting of digits 2 and 4 and multiple of 8 are 224, 424, 244 and 444.

     Of them, only two, namely  "224"  and  "424",  are divisible by 8.

     So, the three right-most digits of the number N are  "224"  or  "424".



(3)  Since the number N is a multiple of 64, it is a multiple of 16, too.

     Hence, its four right-most digits constitute the number, multiple of 16 
            (according to the rule of divisibility by 16).

     In addition, we know that its three right-most digits are  "224"  or  "424".

     The only  possible such four-digit numbers consisting of digits 2 and 4 and multiple of 16 are "2224", "4224", "2424"  or  "4424".

     Of them, only two, namely  "2224"  and  "4224",  are divisible by 16.

     So, the four right-most digits of the number N are "2224"  or  "4224".



(4)  Since the number N is a multiple of 64, it is a multiple of 32, too.

     Hence, its five right-most digits constitute the number, multiple of 32 
            (according to the rule of divisibility by 32).

     In addition, we know that its four right-most digits are  "2224"  or  "4224".

     The only  possible such five-digit numbers consisting of digits 2 and 4 and multiple of 32 are "22224", "42224", "24224"  or  "44224".

     Of them, only two numbers, namely  "24224"  and  "44224",  are divisible by 32.

     So, the five right-most digits of the number N are  "24224"  or  "44224".



(5)  The number N is a multiple of 64.  Hence, it is a multiple of 32, too.
     
     Due to it, we know that its five right-most digits are  "24224"  or  "44224".

     The only  such six-digit numbers consisting of digits 2 and 4 are "224224", "424224", "244224"  or  "444224".

     Of them, only two are divisible by 64:  244224  and  444224.

     So, these two numbers are six-digit numbers satisfying the given condition.



ANSWER.  244224  and  444224 are two six-digit numbers satisfying the given condition.

         These two numbers are UNIQUE: there are no other six-digit integer positive numbers satisfying imposed conditions.