Give a direct proof. If 3|m, then 3|m^2.

[shivers20]

a.) Give a direct proof. If 3|m, then 3|m^2.

Suppose 3|m. Then m=3k for some integer k. Hence m^2=(3k)^2 = 9k^2 =
3(3k^2). Since 3k^2 is an integer, 3|m^2. Done

b.) State the contrapositive of the implication in a.)

If 3|/m, then 3|/m.

c.) Give a direct proof. If 3|/m, then 3|/m^2.

Proof: Suppose 3|/m, then m=3k+1 or m=3k+2 for some integer k.

Case1: Let m=3k+1 for some integer k. So m^2=(3k+1)^2= 3(3k^2+2k)+1
So, 3|/m^2.

Case 2: Let m=3k+2 for some integer k. So m^2=(3k+2)^2= (3k+2)(3k+2)=
9k^2+12k+4= 3(k^2+4k)+4
So, 3|/m^2.

d.) State the contrapostive of the implication in c.)

If 3|m^2, then 3|m.

e.) State the conjunction of the implications in a.) and c.) using "if and only if".

How would I set this up?

* |/ means it is not divisible

 

[pka]

Check b.
The contrapositive is: If 3 \(\displaystyle \not\mid\) m<SUP>2</SUP> then 3 \(\displaystyle \not\mid\)m.

You have c identical to a.

 

[JakeD]

a.) Give a direct proof. If 3|m, then 3|m^2.

Suppose 3|m. Then m=3k for some integer k. Hence m^2=(3k)^2 = 9k^2 =
3(3k^2). Since 3k^2 is an integer, 3|m^2. Done

b.) State the contrapositive of the implication in a.)

If 3|/m, then 3|/m.

c.) Give a direct proof. If 3|/m, then 3|/m^2.

Proof: Suppose 3|/m, then m=3k+1 or m=3k+2 for some integer k.

Case1: Let m=3k+1 for some integer k. So m^2=(3k+1)^2= 3(3k^2+2k)+1
So, 3|/m^2.

Case 2: Let m=3k+2 for some integer k. So m^2=(3k+2)^2= (3k+2)(3k+2)=
9k^2+12k+4= 3(k^2+4k)+4
So, 3|/m^2.

d.) State the contrapostive of the implication in c.)

If 3|m^2, then 3|m.

e.) State the conjunction of the implications in a.) and c.) using "if and only if".

How would I set this up?

* |/ means it is not divisible

For (e), use (a) and (d), or the corrected (b) and (c).