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**JaneFairfax****Member**- Registered: 2007-02-23
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**JaneFairfax****Member**- Registered: 2007-02-23
- Posts: 6,868

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**JaneFairfax****Member**- Registered: 2007-02-23
- Posts: 6,868

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**JaneFairfax****Member**- Registered: 2007-02-23
- Posts: 6,868

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**JaneFairfax****Member**- Registered: 2007-02-23
- Posts: 6,868

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**JaneFairfax****Member**- Registered: 2007-02-23
- Posts: 6,868

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**JaneFairfax****Member**- Registered: 2007-02-23
- Posts: 6,868

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**JaneFairfax****Member**- Registered: 2007-02-23
- Posts: 6,868

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**Muggleton****Member**- Registered: 2009-01-15
- Posts: 64

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**JaneFairfax****Member**- Registered: 2007-02-23
- Posts: 6,868

Yup!

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**JaneFairfax****Member**- Registered: 2007-02-23
- Posts: 6,868

*Last edited by JaneFairfax (2009-03-22 03:33:42)*

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**mathsyperson****Moderator**- Registered: 2005-06-22
- Posts: 4,900

d=e=1 disproves both parts of #9.

(*But not anymore.*)

Why did the vector cross the road?

It wanted to be normal.

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**JaneFairfax****Member**- Registered: 2007-02-23
- Posts: 6,868

I misunderstood the term square free. I thought it was synonymous with non perfect square (and therefore excluded 1).

I have rephrased the question. It shouldnt be ambiguous now.

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**JaneFairfax****Member**- Registered: 2007-02-23
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**JaneFairfax****Member**- Registered: 2007-02-23
- Posts: 6,868

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**Kurre****Member**- Registered: 2006-07-18
- Posts: 280

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**smiyc86****Member**- Registered: 2009-03-19
- Posts: 78

why has no one tried #1 ???

newayz ...

we can rewrite the equn as ... (n-2)(n-1)(n²-2)(n+1)(n+2)

we can clearly see that this expression will be div for n= 7k+1,7k+2,7k+5,7k+6 ... as in each of these casesone of n-1,n-2,n+1,n+2 ... contributes a factor of 7

for 7k+3 and 7k+4 we can find, by substitution, that n²-2 donates a factor of 7.

hence the expression is div by 7 ...

now, if n-1 and n+1 are even and as they are consecutive nos. we can write them as ... 2a,2a+2

now, 2a*(2a+2) = 4a*(a+1)

now a(a+1) will be div by 2! hence div of 8 is also true when n-1 is even

now if n-2 is even ... so is n²-2 and n+2 ....

3 even nos hence three multiples of 8 ... hence div by 8 ....

thus the exp is always div by 56 ...

I love Maths and Music ... dunno which more

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**JaneFairfax****Member**- Registered: 2007-02-23
- Posts: 6,868

@**Kurre**

Well done!

@**smiyc86**

Please use the hide tags.

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**JaneFairfax****Member**- Registered: 2007-02-23
- Posts: 6,868

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**smiyc86****Member**- Registered: 2009-03-19
- Posts: 78

@jane ... how to use the hidden text ??? some html codes reqd if so give the name of the starting tag ... btw i was going o use 7k-1 and 7k-2 instead of 7k+6 and 7k+5 .... but somehow didn't

I love Maths and Music ... dunno which more

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**smiyc86****Member**- Registered: 2009-03-19
- Posts: 78

I love Maths and Music ... dunno which more

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**smiyc86****Member**- Registered: 2009-03-19
- Posts: 78

*Last edited by smiyc86 (2009-03-30 19:52:55)*

I love Maths and Music ... dunno which more

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**smiyc86****Member**- Registered: 2009-03-19
- Posts: 78

*Last edited by smiyc86 (2009-03-31 18:32:37)*

I love Maths and Music ... dunno which more

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**JaneFairfax****Member**- Registered: 2007-02-23
- Posts: 6,868

**smiyc86**:

#6. Brilliant!

#9. Please expand on the proof.

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**smiyc86****Member**- Registered: 2009-03-19
- Posts: 78

@9 ... done

I love Maths and Music ... dunno which more

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