You are not logged in.

- Topics: Active | Unanswered

Pages: **1**

**luca-deltodesco****Member**- Registered: 2006-05-05
- Posts: 1,470

apparently there is some simple proof for this, but i cant find it.

i have to proove that n(n+1)(n+2) where n is a natural number, is always a multiple of 6

The Beginning Of All Things To End.

The End Of All Things To Come.

Offline

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

The expression (call it *A*) is a product of three consecutive integers, and any three consecutive integers always contain a multiple of 3. Therefore *A* is divisible by 3.

It is also divisible by 2 because any three consecutive integers must also contain at least one even number.

Since *A* is divisible by both 2 and 3, and 2 and 3 are coprime, it follows that *A* is divisible by 6.

*Last edited by JaneFairfax (2007-03-19 04:59:43)*

Offline

**luca-deltodesco****Member**- Registered: 2006-05-05
- Posts: 1,470

proof its divisable by 3?

The Beginning Of All Things To End.

The End Of All Things To Come.

Offline

**luca-deltodesco****Member**- Registered: 2006-05-05
- Posts: 1,470

oh right, thats common sense, nevermind

The Beginning Of All Things To End.

The End Of All Things To Come.

Offline

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

Either *n* is divisible by 3 or its not.

If it is, fine. If not, then *n* = 3*k*+1 or *n* = 3*k*−1 for some integer *k*. It then follows that (*n*+2) or (*n*+1) respectively would be divisible by 3.

So one of *n*, (*n*+1), (*n*+2) is always divisible by 3. Hence the product *n*(*n*+1)(*n*+2) is always divisible by 3.

*Last edited by JaneFairfax (2007-03-19 12:00:01)*

Offline

**MathsIsFun****Administrator**- Registered: 2005-01-21
- Posts: 7,664

Or to put it another way, the next "divisible by 3" number is only 3 numbers away, so if you have 3 consecutive numbers, one will be.

Isn't it odd that we can say even, but there is no easy word for "divisible by 3" ... maybe "threven" ? And the two in-between would be "throdds"?

"The physicists defer only to mathematicians, and the mathematicians defer only to God ..." - Leon M. Lederman

Offline

**luca-deltodesco****Member**- Registered: 2006-05-05
- Posts: 1,470

and isn't it odd that you happened to use odd when talking about that, referring to the common pair of odd and even

The Beginning Of All Things To End.

The End Of All Things To Come.

Offline

**MathsIsFun****Administrator**- Registered: 2005-01-21
- Posts: 7,664

I even find it odd.

"The physicists defer only to mathematicians, and the mathematicians defer only to God ..." - Leon M. Lederman

Offline

**Stanley_Marsh****Member**- Registered: 2006-12-13
- Posts: 345

6=2x3,

There are two kinds of integers 2k and 2k+1 , thus it can be divided by 2

There are three kinds of integers 3k , 3k+1, 3k+2 , thus it can be divided by 3

Numbers are the essence of the Universe

Offline

**George,Y****Member**- Registered: 2006-03-12
- Posts: 1,306

If you divide a natural by 3, the remainder is 0, 1, or 2. No other possibilities.

**X'(y-Xβ)=0**

Offline

**George,Y****Member**- Registered: 2006-03-12
- Posts: 1,306

Offline

Pages: **1**