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Algebra

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2 Jul 2008 19:37:16 IST

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Some easy probs

None

Ideally, these are for the 9th and 10th Graders who I am happy to see are quite active in this forum.

If they take more than a day, the seniors may please wade in

1. Problem 1

$a_1, a_2, a_3,...,a_{2000} \in \mathbb{N} \\ \\<br/>\frac{1}{a_1}+\frac{1}{a_2}+\frac{1}{a_3}+...+\frac{1}{a_{2000}} = 1$

Prove that at least one of the numbers has to be even.

2. Problem 2

Prove that $\log_2 3$ is irrational.

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Comments (8)

Rahul Duggal

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2 Jul 2008 20:42:27 IST

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Re:Some easy probs

®µD®A

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2 Jul 2008 21:00:55 IST

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$Let \log_2 3=x\\\\<br/>\Rightarrow2^x=3\\\\\text{Let x is a rational number}\\\\\Rightarrow x= \frac{p}{q}\text{Where p and q are real numbers and (p,q)=1}\\\\2^\frac{p}{q}=3\\\\\Rightarrow2^p=3^q\\\\But\ we\ know\ that\ 2^p\ is\ even\ and\ 3^q\ is\ odd\\\\\text{Hence it is not possible.So our assumption is wrong.And hence x is irrational.}\\\\Hence\ \log_2 3\ is\ irrational.$

Rahul Duggal

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2 Jul 2008 21:09:58 IST

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edited

Rahul Duggal

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Joined: 10 May 2008

Posts: 289

2 Jul 2008 21:12:51 IST

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Re:Some easy probs

Hari Shankar

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3 Jul 2008 09:47:12 IST

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Fantastic work guys. Kudos!

®µD®A

Blazing goIITian

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3 Jul 2008 09:51:17 IST

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As easy as anyone will get. Anyways thanks sir for the appreciation.

®µD®A

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3 Jul 2008 09:56:01 IST

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But sir i know you have another process for the first one.

Hari Shankar

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3 Jul 2008 11:40:32 IST

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The method will simply be a variation of the method given above. This one is fine.

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