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I want to solve this problem be without computer assisted solution.

Note by Chung Kevin 3 years, 1 month ago

Easy Math Editor

*italics*

_italics_

**bold**

__bold__

- bulleted- list

1. numbered2. list

paragraph 1paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)

> This is a quote

This is a quote

# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"

2 \times 3

2^{34}

a_{i-1}

\frac{2}{3}

\sqrt{2}

\sum_{i=1}^3

\sin \theta

\boxed{123}

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Note first that \(a = 512 = 2^{9}, b = 675 = 3^{3}5^{2}\) and \(c = 720 = 2^{4}3^{2}5,\) and thus \(2c^{2} = 3ab.\)

Next, note that

\(a^{3} + b^{3} + c^{3} = a^{3} + b^{3} - c^{3} + 2c^{2}c = a^{3} + b^{3} - c^{3} + 3abc = (a + b - c)(a^{2} + b^{2} + c^{2} - ab + ac + bc).\)

Thus \(a + b - c = 512 + 675 - 720 = 467\) divides the given expression, i.e., the given expression is composite.

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Nice observation with \( 2c^3 = 3ab \)!

512^3-512+675^3-675+720^3-720+(512+675+720) by fermats little theoram it is divisible by 3

Hm, can you explain in detail? I'm pretty sure that the number is not a multiple of 3.

The problem here is the fact that hcf is one. So, I feel that fermats little theorem must be used

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Easy Math Editor

`*italics*`

or`_italics_`

italics`**bold**`

or`__bold__`

boldNote: you must add a full line of space before and after lists for them to show up correctlyparagraph 1

paragraph 2

`[example link](https://brilliant.org)`

`> This is a quote`

Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.`2 \times 3`

`2^{34}`

`a_{i-1}`

`\frac{2}{3}`

`\sqrt{2}`

`\sum_{i=1}^3`

`\sin \theta`

`\boxed{123}`

## Comments

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TopNewestNote first that \(a = 512 = 2^{9}, b = 675 = 3^{3}5^{2}\) and \(c = 720 = 2^{4}3^{2}5,\) and thus \(2c^{2} = 3ab.\)

Next, note that

\(a^{3} + b^{3} + c^{3} = a^{3} + b^{3} - c^{3} + 2c^{2}c = a^{3} + b^{3} - c^{3} + 3abc = (a + b - c)(a^{2} + b^{2} + c^{2} - ab + ac + bc).\)

Thus \(a + b - c = 512 + 675 - 720 = 467\) divides the given expression, i.e., the given expression is composite.

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Nice observation with \( 2c^3 = 3ab \)!

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512^3-512+675^3-675+720^3-720+(512+675+720) by fermats little theoram it is divisible by 3

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Hm, can you explain in detail? I'm pretty sure that the number is not a multiple of 3.

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The problem here is the fact that hcf is one. So, I feel that fermats little theorem must be used

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