(10 points) (10 points) We will now prove the following statement found in Probl

Question

(10 points) (10 points) We will now prove the following statement found in Problem (3): For each integer n that is greater than 1, if a is the smallest positive factor of n that is greater than 1, then a is prime Proof. We will prove the statement using a proof by contradiction. Assume that n is a natural number greater than 1, and that a is the smallest factor of n On the contrary assume that a Thus, there exists a natural number 6 that is a factor of a. Now continue with the proof by finding a contradiction and submit the entire proof in class on Wednesday, Oct 11

Explanation / Answer

Proof: We will prove the statement using a proof of contradiction. Assume that n is a natural number greater than 1 and that a is the smallest factor of n.

On the contrary assume that a is composite. Thus there exists a natural number b that is a factor of a.

Since b is a factor of a, let a = kb where k is a natural number.

Since a is a factor of n, let n = ra, where r is a natural number.

=> n = r(kb)

=> n = (rk)b

=> rk = n/b.

Since r and k are natural numbers, rk is also a natural number.

Therefore n/b is a natural number and b divides n.

Therefore b is a factor of n.

But since b is a factor of a, b < a and b should be the smallest factor of n and not a.

Thus we arrive at a contradiction.

Therefore a is prime.

Related Questions

Navigate

• Browse (All)
• Browse #
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.