Proving the Theorem If m divides n, then m <= n Using Different Methods

[SurferStrobe]

Homework Statement

Prove a theorem using direct proof, mathematical induction, contraposition, or contradiction.

Homework Equations

"If m divides n, then m <= n."

The Attempt at a Solution

(a) Suppose m divides n, then m = nk for some integer k.

(b) If k = 1, the m = n(1) = n.

That show equality part. How do I now show inequality? I'm at a loss for the next step.

 

[HallsofIvy]

If k is not 1, then, since it is a positive integer, k>1. What does that tell you?

 

[SurferStrobe]

Given that, for k=1, m = n,

then for k > 1 (or k + 1),

m < n(k + 1).

If I divide by k + 1,

m / (k + 1) < n.

Am I on the right track?

 

[Sleek]

Homework Statement

Prove a theorem using direct proof, mathematical induction, contraposition, or contradiction.

Homework Equations

"If m divides n, then m <= n."

The Attempt at a Solution

(a) Suppose m divides n, then m = nk for some integer k.

(b) If k = 1, the m = n(1) = n.

That show equality part. How do I now show inequality? I'm at a loss for the next step.

If m divides n (m<=n), then m*k=n and not m=n*k.

If k=1, m=n, if k>1,

mk=n => m<n.

 

[HallsofIvy]

You're right! I missed that completely!

 

[SurferStrobe]

Sleek, Thanks! I guess I got that twisted. Appreciate your helping me understand this logically.

surferstrobe