The proof of 1), for a three-digit number "abc"

abc = 100a + 10b + 1c

= 99a + 9b + (a + b + c)

3 divides 99a + 9b, so 3 divides "abc" iff 3 divides (a + b + c), the sum of the digits.

You can do this for numbers of any length, but I'm not latexing that from my iPhone.

2). Express any number as "abcdef00" + "gh" (for example).

Since "abcdef00" is a multiple of 100, it is divisible by 4. So all that remains is to check "gh", the last two digits.

3) same as 2), since 8 divides 1000

4) same as 1