   Chapter 2.4, Problem 33E

Chapter
Section
Textbook Problem

# Use the fact that 3 is a prime to prove that there do not exist nonzero integers a and b such that a 2 = 3 b 2 . Explain how this proves that 3 is not a rational number.

To determine

To prove: There do not exist non-zero integers a and b such that a2=3b2. Explain how this proves that 3 is not a rational number.

Explanation

Given information:

3 is a prime number

Formula used:

Unique Factorization Theorem:

Every positive integer n either is 1 or can be expressed as a product of prime integers, and this

factorization is unique except for the order of the factors.

Proof:

Let 3 is a prime.

Suppose, there exists a non-zero integer a and b such that a2=3b2

Assume a=p1p2...pt and b=q1q2...qs be prime factorizations of a and b.

Then,

a2=3b2(p1p2...pt)(p1p2...pt)=3(q1q2...qs)(q1q2

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