are these correct?

1.)

Prove that 2 consecutives numbers multiplied are even...

(2n) (2n+1) = 4n^2 + 2n

and since 2n + 2n proves that 2 evens added must be even and both the number in the above answer are even (both divide by 2) so will be even.

2.)

Prove that any 3 consecutive numbers are divisible by 3?

(x)(x+1)(x+2) = 3(x+1)

if the second one is correct then how do you get 3(x+1) from factorising the brackets?

First of all, I assume by "numbers", you mean integers... ?

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1.)

Prove that 2 consecutives numbers multiplied are even...

(2n) (2n+1) = 4n^2 + 2n

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This proves that any number multiplied by 2 and multiplied by the consecutive number to that is even. Big deal!

If you can't come up with a nice, simple, elegant solution, go the brute force approach:

Take any integer n

Case 1:

Assume n is odd (i.e. there is no integer i such that n = 2i)

It follows then that (n+1) is even (i.e. there is an integer j such that (n+1) = 2j)

So, write it out now: (n)(n+1) = (n)(2j) = 2(nj)

Case 2: Assume n is even (and therefore n+1 is odd), and do the same thing as in case 1.

Bam; You're done!

To simplify it, you could just assert that for any two consecutive integers, one of them must be even. And follow it up with proving that an even number multiplied by an odd number is always an even number.

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Prove that any 3 consecutive numbers are divisible by 3?

(x)(x+1)(x+2) = 3(x+1)

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Yes, you want to get it of the form 3(n).

This question is similar to the first. For any three cosecutive integers, one of them must be a multiple of 3. I'll leave it up to you, if you need to prove it.

Case 1:

Assume x is a multiple of 3; x = 3i

Then...

(x)(x+1)(x+2) = 3i(x+1)(x+2) = (3ix + 3)(x+2) = 3ix^2 + 6ix + 3x + 6

= 3(ix^2 + 2ix + x +2)

Case 2:

Assume (x+1) is a multiple of 3; (x+1) = 3j

Then...

(x)(x+1)(x+2) = (x)(3j)(x+2) = (3jx)(x+2) = 3jx^2 + 6jx

= 3(jx^2 + 2jx)

Case 3:

Assume (x+2) is a multple of 3; (x+2) = 3k

Then...

(x)(x+1)(x+2) = (x)(x+1)(3k) = (x^2 + x)(3k) = 3kx^2 + 3kx

= 3(kx^2 + kx)

Voila! You're done.

So, you can see where this is headed.... How about multiplying 4 consecutive integers? Would it be a multiple of 4? Try and solve it using a similar method.

Like I said, if you can't come up with a simple solution, or using an advanced method (such as moduli), then "brute force" is the way to go. I was just trying to help the guy out.

As for

(2n) (2n+1) = 4n^2 + 2n ...

Yes, I have seen this form before, but it isn't really accurate, as not every integer n can be expressed in the form (2i). It's trivial, but true. For it to work, you have to assert that the first integer in your sequence is even. And then we're not dealing with ANY sequence of two integer, only sequences that start with even numbers.

Nit-pick, nit-pick!