# 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 .

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 .

It certainly looks as if we've got to put $p = 3$ and $q = 2$ here. This gives $3^{2n}-4n - 1$ is divisible by $4$.

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

Can you see what to do next?

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

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

Can you see what to do next?

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

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

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