19845k = 3*3*3*3*5*7*7k = 34*51*72k

by a value that will make it
a perfect cube, we much choose a value that will cause
every factor to have an exponent which is a multiple of 3.

The 34 must be multiplied by 32 so it
will become 36 

The 51 must be multiplied by 52 so it
will become 53

The 72 must be multiplied by 71 so it
will become 73


So we must multiply 34*51*72 by
32*52*71

So k = 32*52*71 = 9*25*7 = 1575

So the smallest possible positive integer value of k is 1575

Checking:

19845*1575 = 31255875 = 315³ 

Edwin