Explanation Given information: Mathematical induction and Euclid’s lemma. Concept used: Mathematical Induction: This method has two steps to be followed: Basis Step Inductive Step. Proof: Basis Step: Prove the statement for the first possible value s = 1 . For s = 1 The statement becomes, “if p and q 1 are prime numbers and p | q 1 then p = q 1 ” By definition of a prime number, it has only two factors − itself and 1 . Since, q 1 is a prime number so it is divisible by 1 and q 1 only. But it is divisible by p . So, p has to be either 1 or q 1 . It is given that p is a prime number so, it can’t be 1 . So, p = q 1 . Therefore, the statement holds true for s = 1 . Inductive Step: Assume that the statement holds true for s = m . Therefore, it is assumed that if p and q 1 , q 2 , .... , q m are prime numbers and p | q 1 , q 2 , q 3 , ...... , q m then p = q i for some i with 1 < i < m is true ........ ( 1 ) . The next case in the series is m + 1 . The following statement has to be proved as true. If p and q 1 , q 2 , ..... , q m , q m + 1 are prime numbers and p | q 1 , q 2 , ..... , q m , q m + 1 then p = q i for some i with 1 ≤ i ≤ m + 1 To determine (b) To prove: The uniqueness part of the unique factorization theorem for integers.

5th Edition

ISBN: 9780357097717

Author: EPP

Publisher: CENGAGE L

See similar textbooks

Concept explainers

Statements and Logic

Logic is the analysis of general patterns of thoughts without regard to any specific sense of context.

Videos

Question

Chapter 8.4, Problem 41ES

To determine

(a)

To prove:

For all positive integers s, if p and q1,q2,...,qs are prime numbers and p|q1q2....qs then p=qi for some i with 1≤i≤s, using mathematical induction and with Euclid’s lemma

To determine

(b)

To prove:

The uniqueness part of the unique factorization theorem for integers.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 8.4, Problem 40ES

Chapter 8.4, Problem 42ES

Students have asked these similar questions

18:36 G.C.A.2.ChordsSecantsandTa... จ 76 完成 2 In the accompanying diagram, AABC is inscribed in circle O, AP bisects BAC, PBD is tangent to circle O at B, and mZACB:m/CAB:m/ABC= 4:3:2 D B P F Find: mZABC, mBF, m/BEP, m/P, m/PBC ← 1 Ő

14:09 2/16 jmap.org 5G 66 In the accompanying diagram of circle O, diameters BD and AE, secants PAB and PDC, and chords BC and AD are drawn; mAD = 40; and mDC = 80. B E Find: mAB, m/BCD, m/BOE, m/P, m/PAD ← G.C.A.2.ChordsSecantsand Tangent s19.pdf (538 KB) + 4 保存... X

16:39 < 文字 15:28 |美图秀秀保存 59% 5G 46 照片完成 Bonvicino - Period Name: 6. A right regular hexagonal pyramid with the top removed (as shown in Diagram 1) in such a manner that the top base is parallel to the base of the pyramid resulting in what is shown in Diagram 2. A wedge (from the center) is then removed from this solid as shown in Diagram 3. 30 Diogram 1 Diegrom 2. Diagram 3. If the height of the solid in Diagrams 2 and 3 is the height of the original pyramid, the radius of the base of the pyramid is 10 cm and each lateral edge of the solid in Diagram 3 is 12 cm, find the exact volume of the solid in Diagram 3, measured in cubic meters. Show all work. (T 文字贴纸消除笔涂鸦笔边框马赛克去美容

Chapter 8 Solutions

DISCRETE MATHEMATICS WITH APPLICATION (

Ch. 8.1 - If R is a relation from A to B, xA , and yB , the...Ch. 8.1 - Prob. 2TY Ch. 8.1 - Prob. 3TY Ch. 8.1 - Prob. 4TY Ch. 8.1 - If R is a relation on a set A, the directed graph...Ch. 8.1 - As in Example 8.1.2, the congruence modulo 2...Ch. 8.1 - Prove that for all integers m and n,m-n is even...Ch. 8.1 - The congruence modulo 3 relation, T, is defined...Ch. 8.1 - Define a relation P on Z as follows: For every...Ch. 8.1 - Prob. 5ES

Knowledge Booster

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, subject and related others by exploring similar questions and additional content below.

(a) To prove: For all positive integers s , if p and q 1 , q 2 , ... , q s are prime numbers and p | q 1 q 2 .... q s then p = q i for some i with 1 ≤ i ≤ s , using mathematical induction and with Euclid’s lemma

Solution Summary: The author explains mathematical induction and Euclid's lemma.

Concept explainers

Videos

(a)

To prove:

For all positive integers s, if p and q1,q2,...,qs are prime numbers and p|q1q2....qs then p=qi for some i with 1≤i≤s, using mathematical induction and with Euclid’s lemma

(b)

To prove:

The uniqueness part of the unique factorization theorem for integers.

Want to see the full answer?

Chapter 8 Solutions

(a) To prove: For all positive integers s , if p and q 1 , q 2 , ... , q s are prime numbers and p | q 1 q 2 .... q s then p = q i for some i with 1 ≤ i ≤ s , using mathematical induction and with Euclid’s lemma

Solution Summary: The author explains mathematical induction and Euclid's lemma.

Concept explainers

Videos

(a) To prove: For all positive integers s, if p and q1,q2,...,qs are prime numbers and p|q1q2....qs then p=qi for some i with 1≤i≤s, using mathematical induction and with Euclid’s lemma

(b) To prove: The uniqueness part of the unique factorization theorem for integers.

Want to see the full answer?

Chapter 8 Solutions

(a)

To prove:

For all positive integers s, if p and q1,q2,...,qs are prime numbers and p|q1q2....qs then p=qi for some i with 1≤i≤s, using mathematical induction and with Euclid’s lemma

(b)

To prove:

The uniqueness part of the unique factorization theorem for integers.