My New Mobile Number

Possible $6$ digit combinations (assuming that digits can be repeated) $= { 10 }^{ 6 }$

So our list is $1,2,3\dots { 10 }^{ 6 }$

Now we need to separate the numbers which are divisible by $1729$ from the list. To do that, we need to simplify the list. First multiple of $1729$ in the list is $1729$ itself while the last multiple is

$\left\lfloor \frac { { 10 }^{ 6 } }{ 1729 } \right\rfloor = 578$

So our new list is $1\cdot 1729,2\cdot 1729,3\cdot 1729\dots 578\cdot 1729$

Since now we just need to count the number of items in the list, we can simply out list as $1,2,3\dots 578$

Therefore, there are $578$ numbers which satisfy the given conditions and hence these numbers are possible candidates for the actual number. Since actual number is hidden within them, when you try all the $578$ numbers, you are guaranteed to get the right answer, even in worst case scenario!