Prove or disprove. if a|b and b|a, where a and b are integers, then a = b. Prove or disprove. if a|bc, where a,b and c are positive integers and a...

1. Prove or disprove. if a|b and b|a, where a and b are integers, then a = b.

2. Prove or disprove. if a|bc, where a,b and c are positive integers and a 6= 0, then a|b or a|c. 

3. Prove or disprove. The product of any two even integers is a multiple of 4. 

4. Prove or disprove. For all integers a,b and c, if a|b and a|c, then a|(2b −3c)

5. Suppose that a and b are integers, a ≡ 4( mod 13), and b ≡ 9( mod 13).

Find the integer c with 0 ≤ c ≤ 12 such that c ≡ 9( mod 13).

6. Evaluate 155 mod 19 and −221 mod 23

7. Find a mod m and a d i v m when

(a) a = 9009,m = 223

(b) a = −10101,m = 333

8. Show that if n is an integer, then n 2 ≡ 0( mod 4) or n 2 ≡ 1( mod 4)