How do I prove that any even number, divided by 2 a number of times, would eventually go down to an odd number or 1?
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1Welcome to MSE. Your question is phrased as an isolated problem, without any further information or context. This does not match many users' quality standards, so it may attract downvotes, or be closed. To prevent that, please [edit] the question. This will help you recognize and resolve the issues. Concretely: please provide context, and include your work and thoughts on the problem. These changes can help in formulating more appropriate answer – José Carlos Santos Feb 27 '24 at 13:15
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1Strong induction, noting that $\frac{1}{2} n < n$ – JMoravitz Feb 27 '24 at 13:20
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2"An odd number or 1" is redundant. Since 1 is an odd number, it suffices to say "An odd number" – Arthur Feb 27 '24 at 13:22
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1An odd question... – Jean Marie Feb 27 '24 at 13:33
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Hope this helps https://math.stackexchange.com/a/25919/1277478 – Vinay Karthik Feb 27 '24 at 13:58
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Let $n$ be an even number. By the Fundamental Theorem of Arithmetic, we have that $$n = 2^{m}\cdot p_{1}^{m_{1}} \cdots p_{k}^{m_{k}},$$ for some prime numbers $p_{i}$, with $p_{i} \not = p_{j}$ for all $i \not = j$, and $2 \not = p_{j}$ for all $j$, and $m, \, m_{j} \geq 0$. Therefore, $\frac{n}{2^{m}}$ is an odd number. In other words, if we divide $n$ by $2$ $m$ times turn into an odd number.
J. W. Tanner
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Please strive not to post more (dupe) answers to dupes of FAQs. This is enforced site policy, see here. It's best for site health to delete this answer (which also minimizes community time wasted on dupe processing.) Using FTA is overkill here. – Bill Dubuque Feb 27 '24 at 14:32