Next, for the powers of 2, over the ring Z/1000Z, all the 100 elements for the form

8 (mod 1000),

40 (mod 1000) are being generated.

Unlike this, for the powers of 3, although over the ring Z/100Z

all the 20 elements

1, 3, 7, 9 (mod 20) are being generated, this does not hold out over on from the insider for the larger ring Z/1000Z at all.

For this example, the element 3, over on multiplication yields, generating the elements 1, 3, 9 (mod 1000), although the element 7 (mod 1000) is not being generated at all, although, instead it rather generates the element 507 (mod 1000), although, rather.

AGAIN WHY?

What is being it to be the true reason behind this, rather?

WHY?