Last sentence should be -891. Forgot the - sign

For the second line, the last term's power should be 999 rather than 1000.

Also, strictly speaking ,

999 mod 1000 = 999
9999 mod 1000 = 9999, and not -1.

Perhaps you meant:

999 ≡ -1 (mod 1000) and 9999 ≡ -1 (mod 1000)


@Nancy : Alternative style of presentation in the last two steps

Based on the multiplication property:

If a ≡ b (mod m) and c ≡ d (mod m), then ac = bd (mod m)

Then 999 ≡ 9999 ≡ 99999 ≡ ... ≡ (999 '9's) ≡ -1 (mod 1000)

Then 999 × 9999 × 99999 × ... × (999 '9's) ≡ (-1)(-1)(-1)... (mod 1000) = (-1)^997 (mod 1000) = -1 (mod 1000)

And since 891 ≡ -109 (mod 1000),

Then 9 × 99 × 999... × (999 '9's)

= 891 × 999... × (999 '9's)

≡ -109 × (-1) (mod 1000) = 109 (mod 1000)