# Thread: Bimonial theorem and expansion

1. ## Bimonial theorem and expansion

Two positive integers , p and q are connected by p=q+1 . By using the binomial expansion , show that the expression p^{2n}-2nq-1 can be divided exactly by q^2 for all positive integers n .

I can do this part .

This is the continuation :

hence(2) show that 3^{15}+5 can be divided exactly by 4 .

i am not sure about this part .

2. ## Divisibility

Hello thereddevils
Originally Posted by thereddevils
Two positive integers , p and q are connected by p=q+1 . By using the binomial expansion , show that the expression p^{2n}-2nq-1 can be divided exactly by q^2 for all positive integers n .

I can do this part .

This is the continuation :

hence(2) show that 3^{15}+5 can be divided exactly by 4 .

i am not sure about this part .
It certainly looks as if we've got to put $\displaystyle p = 3$ and $\displaystyle q = 2$ here. This gives $\displaystyle 3^{2n}-4n - 1$ is divisible by $\displaystyle 4$.

Obviously, we can't put $\displaystyle n = 7.5$, to give $\displaystyle 3^{15}$ directly. But what about $\displaystyle n = 8$? This gives $\displaystyle 3^{16} - 33$ is divisible by $\displaystyle 4$. And $\displaystyle 33$ has $\displaystyle 3$ as a factor ...

Can you see what to do next?

Grandad

3. Originally Posted by Grandad
Hello thereddevilsIt certainly looks as if we've got to put $\displaystyle p = 3$ and $\displaystyle q = 2$ here. This gives $\displaystyle 3^{2n}-4n - 1$ is divisible by $\displaystyle 4$.

Obviously, we can't put $\displaystyle n = 7.5$, to give $\displaystyle 3^{15}$ directly. But what about $\displaystyle n = 8$? This gives $\displaystyle 3^{16} - 33$ is divisible by $\displaystyle 4$. And $\displaystyle 33$ has $\displaystyle 3$ as a factor ...

Can you see what to do next?

Grandad

Thanks Grandad . I know that $\displaystyle 3^{16}-33$ is divisible by 4 but i still can't get it . Really sorry bout that . I guess i need more explaination .

4. ## Divisibility

Hello thereddevils
Originally Posted by thereddevils
Thanks Grandad . I know that $\displaystyle 3^{16}-33$ is divisible by 4 but i still can't get it . Really sorry bout that . I guess i need more explaination .
$\displaystyle 3^{16} - 33 = 3(3^{15} - 11)$, which is divisible by $\displaystyle 4$.

So, since $\displaystyle 3$ isn't divisible by $\displaystyle 4$, $\displaystyle 3^{15}- 11$ is divisible by $\displaystyle 4$

$\displaystyle \Rightarrow 3^{15} - 11 + 16$ is divisible by $\displaystyle 4$

Grandad