8n divides (4n) Proof by induction

[kathrynag]

Homework Statement

I need to do this by proof by induction
8n divides (4n)!

Homework Equations

The Attempt at a Solution

I already did it for 1, now I need to do for k and k+1
(4k)!t=8k

 

[mutton]

If 8k divides (4k)!, how can 8k be a multiple of (4k)! as you wrote?

 

[kathrynag]

$$8^{k}$$ divides (4k)!
$$8^{k}$$*t=(4k)!

 

[mutton]

Simplify (4(k + 1))! in terms of (4k)!

 

[kathrynag]

(4k+4)!=(4k+4)(4k+3)(4k+2)(4k+1)(4k)!

 

[mutton]

You already know that $$8^k$$ divides (4k)!, so does 8 divide everything else?

 

[kathrynag]

Yes...

 

[mutton]

Then you have all the details for the complete proof.

 

[kathrynag]

Ok, I see...