Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

The problem below is from Cupillari's Nuts and Bolts of Proofs.

Prove the following statement:

Let $a$ and $b$ be two relatively prime numbers. If there exists an $m$ such that $(a/b)^m$ is an integer, then $b=1$.

My question is: Is the statement true?

I believe the statement is false because there exists an $m$ such that $(a/b)^m$ is an integer, and yet $b$ does not have to be $1$. For example, let $m=0$. In this case, $(a/b)^0=1$ is an integer as long as $b \neq 0$.

So I think the statement is false, but I am confused because the solution at the back of the book provides a proof that the statement is true.

share|improve this question
6  
Presumably the problem meant that $m>0$ and $m$ is an integer. It's not true if $m<0$ either, since $(\frac{1}{2})^{-1}$ is an integer. –  Thomas Andrews Nov 8 '11 at 19:28
    
(Actually, it's true for rational $m>0$, not just integer $m$.) –  Thomas Andrews Nov 8 '11 at 19:31
    
@thomas Thank you! This gives me some confidence to fill in gaps in other incomplete problem statements I might encounter. –  Andrew Liu Nov 8 '11 at 19:39
    
I think Cupillari should have said "positive integer". –  Michael Hardy Nov 8 '11 at 20:21

2 Answers 2

up vote 3 down vote accepted

Your counterexample is valid. But the statement is true if $m$ is required to be a positive natural number or positive integer.

Alternatively, note it's not if true $m$ is required to be negative.

In my opinion, it seems like you were supposed to assume $m>0$.

share|improve this answer

This statement is true under certain conditions. You must assume $b \neq 0$. If $(a,b)=1$, you can easily show that $(a^n, b^n) = 1$. This means that $(a/b)^n = a^n / b^n$ is never an integer unless there exists $n$ such that $b^n = \pm 1$, but that means $b = \pm 1$.

The case where $a = 0$ and $|b| > 1$ must be excluded because then $(a,b) > 1$.

Hope that helps,

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.