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Elements Of Modern Algebra

8th Edition
Gilbert + 2 others
ISBN: 9781285463230
BuyFindarrow_forward

Elements Of Modern Algebra

8th Edition
Gilbert + 2 others
ISBN: 9781285463230
Textbook Problem

a. The binomial coefficients ( n r ) are defined in Exercise 25 of Section 2.2 . Use

induction on r to prove that if p is a prime integer, then p is a factor of ( p r ) for r  = 1 ,   2 ,   .   .   .   ,   p 1 . (From Exercise 26 of Section 2.2 , it is known that ( p r ) is an integer.)

b. Use induction on n to prove that if p is a prime integer, then p is a factor of n p n .

(a)

To determine

To prove: If p is a prime integer, then p is a factor of (pr) for r=1,2,,p1.

Explanation

Given information:

Use induction on r. (pr) is an integer.

Formula used:

1) Mathematical Induction:

The given statement Pn is true for all positive integers n if,

a. Pn is true for n=1

b. The truth of Pk always implies that Pk+1 is true.

2) Binomial Theorem:

(a+b)n=(n0)an+(n1)an1b+(n2)an2b2++(nr)anrbr++(nn)bn.

Where the binomial coefficients (nr) are defined by (nr)=n!(nr)!r!

With r!=r(r1)(2)(1) for r1 and 0!=1.

Proof:

Let p is a prime integer.

Consider the statement p is a factor of (pr) for r=1,2,,p1.

By mathematical induction,

a. For r=1,

Since, (p1)=p!(p1)!1!

=p(p1)!(p1)!

=p

Since, p is a divisor of p.

Thus, p is a factor of (pr) for r=1.

Therefore, the statement is true for r=1.

b. Assume that the statement is true for r1.

That is, p is a factor of (pr1) for some 2rp1.

p|(pr1).

To show the statement is true for r

(b)

To determine

To prove: If p is a prime integer, the p is a factor of npn.

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