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These are few questions I encountered while self teaching some elementary number theory, I've answered some but I don't have any solutions to check If I'm right.

  1. If $a \mid m$ and $b \mid n$, prove that $ab \mid mn$.

  2. Does $6 \mid 11n$ imply that $6 \mid n$? Justify. Does $6 \mid 34n$ imply $6 \mid n$? Justify.

  3. If $p$ is a prime, prove that $(p, a) = 1$ or $(p, a) = p$.

  4. If a is a positive integer, what is $(a, 2a)$? What is $(a, a2)$?, $(a, a + 1)$?, and $(a, a + 2)$?

  5. Show that the difference of two consecutive cubes is never divisible by $3$.

  6. Show that if $(a, b) = 1$, then $(a + b, a - b) = 1$ or $2$.

  7. Prove that square of an integer is either a multiple of $4$ or one more than a multiple of $4$.

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4  
But where are your solutions to those you have solved so we can check them? And where are you stuck on the others? –  Tobias Kildetoft Aug 1 '13 at 7:40
    
Why don't you provide your answers to the questions you could do, so you can get feedback on your work? –  Zev Chonoles Aug 1 '13 at 7:41
    
I'm stuck with the last two. As for the solutions for the others, I'm on my phone so I can't really type them now. –  Mohamed Ayman Aug 1 '13 at 7:42

1 Answer 1

Here are my thoughts. If you disagree, I'll expand my answers.

  1. We have $m = ax$ and $n = by$. Therefore $mn=(ax)(by)=ab(xy)$. Hence, $ab|mn$.
  2. Yes (ask for proof). No consider $n=3$.
  3. Only divisors of $p$ are by definition $1$ and $p$, so its gcd with another number could only possibly be one of these two.
  4. $(a,2a)=1$. Not sure what you mean with this. $(a,a+1)=1$. $(a,a+2)=1 \text{ or } 2$.
  5. $(n+1)^3 - n^3 = 3n^2 + 3n + 1 = 3(n+n^2) + 1$.
  6. If $d=(a+b,a-b)$, then $d \mid a+b$ and $d \mid a-b$. Therefore, $d|(a+b)-(a-b) \iff d \mid 2a$. Also, $d|(a+b)+(a-b) \iff d \mid 2b$. Because $(a,b)=1$, $d$ must be either $1$ or $2$.
  7. Either $n=2k$ or $n=2k+1$. In first case, $n^2 = (2k)^2 = 4k^2$. In second, we have that $n^2 = (2k+1)^2 = 4k^2 + 4k + 1 = 4(k^2 + k) + 1$.
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I don't know why somebody down voted your answer, anyway I like your solutions. Thank you. –  Mohamed Ayman Aug 1 '13 at 8:15
    
@MohamedAyman If your having trouble, you may want to look at Underwood Dudley's Elementary Number Theory. The first edition is free on archive.org. The second is about $10 on amazon or free in a library! Answers and comments on select problems are in the back. –  Gamma Function Aug 1 '13 at 8:17

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