We break 1440 down into its prime factorization:

1440 = 2×720 = 2×2×360 = 2×2×2×180 = 2×2×2×2×90 = 2×2×2×2×2×45 =

2×2×2×2×2×3×15 = 2×2×2×2×2×3×3×5 = 25×32×51

The smallest cube larger than that is the smallest one in which
the exponents of the primes 2,3, and 5 are all multiples of 3.
That would be 26×33×53.

So 
we need 21 to multiply by the 25 to give 26
we need 31 to multiply by the 32 to give 33
we need 52 to multiply by the 51 to give 53

So we need to multiply 1440 by

21×31×52 = 150 so that it will become

26×33×53, and have all the exponents of
prime numbers being multiples of 3.

The sum of the digits of 150 is 1+5+0 = 6.

Edwin