Definition: x | y (x divides y)
For x ∈ Z, x ≠ 0; y ∈ Z
x | y ⇔ ∃z ∈ Z such that xz = y
1. If a | b, then a | bc for all c (where c is an integer)
Prof. Conrad solved this one in class on Monday, but here is a more clear version.
We are trying to prove a | bc
That means
we are looking to prove ∃j∈ Z such that ja= bc
To find it, we convert the statement a|b into an equation by appying the definition. We work with that equation until we get one that is in the form we need, that is some integer j times a on the left, and bc on the right. This same techinque can be applied to the three other problems below as well.
| step number | step | justification |
| (1) | a|b | given |
| (2) | ∃k∈ Z such that ak=b | definition of divides (|), using ⇒ |
| (3) | ∀c∈Z, akc = bc | algebra (multiply both sides by c) |
| (4) | Let j=kc | defining a variable |
| (5) | ∃j∈ Z such that ja=bc | combine steps 3 and 4 into the form of the right hand side of the definition of divides |
| (6) | a|bc | defintion of divides (|), using ⇐ |
Now, you try to prove these three theorems from the notes in class on Monday
2. If a | b and b | c, then a | c
3. If a | b and a | c, then a | sb + tc for all s and t.
4. For all c ≠ 0, a | b if and only if ca | cb.
P. Conrad, 04/14/2008