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#1 2007-05-25 08:40:48

Daniel123
Member
Registered: 2007-05-23
Posts: 663

Proving - squares

I can do the first part, but cannot do the second part. If you are going to help me, could you please also explain how/why you thought/decided to do what you did for part b. thanks!

a) by expanding both sides,
show (m²+1)(n²+1) = (m+n)² + (mn-1)²

  ∴ they are equal

b) Using this result, write 500,050 as a sum of 2 square numbers

Thank you!!

Last edited by Daniel123 (2007-05-25 08:50:33)

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#2 2007-05-25 09:14:15

mathsyperson
Moderator
Registered: 2005-06-22
Posts: 4,900

Re: Proving - squares

Well, the question wants us to write it as a sum of two squares, and that fits the right-hand side of the equation in the first part.

So to use the result we need to find a way of writing 500,050 in the form of the left-hand side, which is two numbers multiplied together such that each number is one more than a square.

Just by looking at the number, 50 seems like it would be a good candidate. This is 7² + 1, so it fits the condition we want. If 50 is the first number, then the other would have to be 10001.
That one is also good, because that it 100² + 1.

So now we have 500050 = (100²+1)(7²+1).

By using the result from a), that means that 500050 = 107² + 699².
Checking with a calculator confirms this.


Why did the vector cross the road?
It wanted to be normal.

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#3 2007-05-25 09:21:50

Daniel123
Member
Registered: 2007-05-23
Posts: 663

Re: Proving - squares

Thank you! that was really well explained. Ive realised the problem i have is deciding how to go about solving it - once you said to look at the LHS and find two numbers that multiply to give 500,050 i could do the rest myself. So ive learnt... if i see 'using this result', make sure i do actually use it! Thanks smile

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#4 2007-05-25 09:55:45

mathsyperson
Moderator
Registered: 2005-06-22
Posts: 4,900

Re: Proving - squares

Yep, pretty much.

b) on its own would be a horrific nightmare of a question, and I'd probably have to resort to brute-forcing it in Excel. If I got that it an exam or something then I'd have no idea what to do.

But as a) came before it, it teaches you a nice trick to make the problem a lot simpler. It's worth looking to see if previous results from a question can be used even if it doesn't explicitly tell you to.

Edit: Now I think about it, that would be an interesting way of generating Pythagorean Triples.


Why did the vector cross the road?
It wanted to be normal.

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