Divisibility and Congruence Discussion

(The following is/are the initial prompt(s) for an online discussion; students may have posted responses, and prompts for further discussion may have been added, but these things are not shown.)

This discussion helps you understand divisibility and congruence, and how to construct direct proofs involving them.

The Ideas

Determine whether the following claims are true or false. Explain your decisions in terms of the definitions of divisibility  or congruence, as appropriate.

1. 4 | 20
2. 4 | -8
3. 4 | 13
4. -4 | 12
5. 6 | 20
6. 4 ≡ 13 (mod 3)
7. 4 ≡ 13 (mod 6)
8. 4 ≡ 20 (mod 3)
9. 6 ≡ 4 (mod 2)
10. -5 ≡ 0 (mod 5)
11. -8 ≡ -23 (mod 5)

Proofs

(From Progress Check 3.2 in our textbook.) See if you can prove the following: If a, b, and c are integers such that a|b and a|c, then a|(b+c).

Can you prove the following: If a and b are integers such that a ≡ 2 (mod 3) and b ≡ 2 (mod 3), then (a+b) ≡ 1 (mod 3).

Does the second proof above remind you of anything from problem set 2?