Prove that if k and n are positive integers and k divides n, then $\displaystyle 2^k-1$ divides $\displaystyle 2^n-1$. I feel like I've done this problem before, but I'm not sure how to approach it.
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Originally Posted by jmedsy Prove that if k and n are positive integers and k divides n, then $\displaystyle 2^k-1$ divides $\displaystyle 2^n-1$. I feel like I've done this problem before, but I'm not sure how to approach it. Since k divides n, n = kl for some positive integer l. Use the fact that $\displaystyle x-1$ divides $\displaystyle x^m - 1$ for any positive integer $\displaystyle m$ Choose $\displaystyle x = 2^k$ and $\displaystyle m = l$ in the above statement to get your answer.
Originally Posted by Isomorphism Since k divides n, n = kl for some positive integer l. Use the fact that $\displaystyle x-1$ divides $\displaystyle x^m - 1$ for any positive integer $\displaystyle m$ Choose $\displaystyle x = 2^k$ and $\displaystyle m = l$ in the above statement to get your answer. I understand, thanks.
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