Math Is Fun Forum

bubokribuck

Member

Registered: 2011-10-28

Posts: 42

Prove k^4+2k^3+2k+k is divisible by 6.

The question is to prove by induction that

is divisible by 30 (n is any positive integer). I've done up to the stage where I have

As I've already assumed that k^5-k is true (i.e, it's divisible by 30), so I now just need to prove that

is divisible by 6 so that

would be divisible by 30 too, and then I'll have solved the problem.

However, I am struggled to prove that

is divisible by 6. Can someone help me please.

Last edited by bubokribuck (2011-11-01 15:28:29)

TheDude

Member

Registered: 2007-10-23

Posts: 361

Re: Prove k^4+2k^3+2k+k is divisible by 6.

If a number is divisible by 2 and 3 then it is divisible by 6, so we need to prove that k^4 + 2k^3 + 2k^2 + k is divisible by 2 and 3.  Proving it for 2 is quick: 2k^3 + 2k^2 is clearly divisible by 2.  For k^4 + k, note that if k is even then k^4 is even and if k is odd then k^4 is odd.  So k^4 + k is either even + even or odd + odd, both of which result in an even number, which is of course is divisible by 2.

For 3, go back to the assumption.  We assume that k^5 - k is divisible by 30, which means it is also divisible by 3.  Equivalently, k(k^4 - 1) is divisible by 3.  That means that either k or k^4 - 1 is divisible by 3.  If k is divisible by 3 then k^4 + 2k^3 + 2k^2 + k clearly is as well.  If not then either k = 3x + 1 or k = 3x + 2 where x is an integer.  Make both substitutions and prove that the result is divisible by 3, then you're done.

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Wrap it in bacon

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Prove k^4+2k^3+2k+k is divisible by 6.

so I now just need to prove that

is divisible by 6

When k = 2 then k^4+2k^3+2k+k = 38 which is not divisible by 6

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In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

TheDude

Member

Registered: 2007-10-23

Posts: 361

Re: Prove k^4+2k^3+2k+k is divisible by 6.

It's a typo, he meant k^4 + 2k^3 + 2k^2 + k.

Last edited by TheDude (2011-11-02 07:01:47)

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Wrap it in bacon

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Prove k^4+2k^3+2k+k is divisible by 6.

That makes it a lot better. Now you you can prove that

like this:

Factor:

All the k are of 3 types. 0 mod 3, 1 mod 3 or 2 mod 3.

If k is 0 mod 3 then it is divisible by 3 and

is obviously divisible by 3 because k is.

If k is 2 mod 3 then

is divisible by 3 because the term ( k + 1 ) is.

If k is 1 mod 3 then

is divisible by 3 because the term ( k^2 + k + 1 ) is.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

bubokribuck

Member

Registered: 2011-10-28

Posts: 42

Re: Prove k^4+2k^3+2k+k is divisible by 6.

Thanks for TheDude correcting my error, and thank you both for helping me solve the problem, I have successfully solved it. Thanks

Last edited by bubokribuck (2011-11-02 10:24:24)