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Maths proof question

So this question was in my iGCSE exam and I'm not sure how it would be solved.
SHOW that 2^(p+1) is a factor of (2^p-1)^2-1

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Reply 1

RogerOxon

Here's the start:

(2^p-1)^2-1=2^{2p}-2.2^p+1-1

Continue to simplify.

(edited 6 years ago)

Reply 2

Futurechemist

Original post by RogerOxon

Here's the start:

(2^p-1)^2-1=2^{2p}-2.2^p+1-1

Continue to simplify.

Then how do you show ?

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Reply 3

RDKGames

Study Forum Helper

Original post by Futurechemist

Then how do you show ?

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By factoring out

2^{p+1}

Reply 4

Futurechemist

Original post by RDKGames

By factoring out

2^{p+1}

Can you please show me how it's done

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Reply 5

RDKGames

Study Forum Helper

Original post by Futurechemist

Can you please show me how it's done

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(2^p-1)^2-1=(2^p)^2-2\cdot2^p+1-1=2^{2p}-2^{p+1}

We also have

2^{2p}=2^{(p-1)+(p+1)}=2^{p-1}\cdot 2^{p+1}

So you have

(2^p-1)^2-1=2^{p-1}\cdot 2^{p+1}-2^{p+1}

and you can finish it off from there

Reply 6

RogerOxon

Original post by RogerOxon

Here's the start:

(2^p-1)^2-1=2^{2p}-2.2^p+1-1

Continue to simplify.

(2^p-1)^2-1=2^{2p}-2.2^p+1-1=2^p(2^p-2)=2^p2(2^{p-1}-1)=?

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Maths proof question

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