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

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The theorem states that if m divides n, then m must be less than or equal to n. The proof begins by expressing m as nk for some integer k. If k equals 1, then m equals n, confirming equality. For k greater than 1, it follows that m is less than n, as m equals n divided by k, which is a positive integer greater than 1. The discussion emphasizes the logical steps needed to establish the inequality, clarifying the relationship between m and n based on the value of k.
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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.
 
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If k is not 1, then, since it is a positive integer, k>1. What does that tell you?
 
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?
 
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SurferStrobe said:

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.
 
You're right! I missed that completely!
 
Sleek, Thanks! I guess I got that twisted. Appreciate your helping me understand this logically.

surferstrobe
 

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