Prove algebraically - odd numbers

Daniel123 · 2007-05-23 23:48:13

Im doing past gcse papers, and I came across this question:

Prove algebraically that the sum of the squares of any two odd numbers leaves a remainder of 2 when divided by 4.

I have tried this, and i think i am on the right lines, but i have two problems with it. My workings so far:

An odd number can be represented by 2r+1, and 2r-1 etc.

(using question and dividing by 4)

This has not left a remainder of 2, but a remainder of 1/2 instead. However, I thought that when dividing by 4, a remainder of 2 would mean a remainder of 2/4 = 1/2, so the 1/2 is a remainder of 2.

But if this remainder is correct, Im still not sure I have proved it. (2r+1) and (2r-1) are both odd numbers, but they are consecutive odd numbers - so would using them be the same as using non-consecutive odd numbers? So have i only proved that that the sum of the square of two consecutive odd numbers gives a remainder of 2 when divided by 4? If so how would i go about doing the question?

Thanks.

luca-deltodesco · 2007-05-23 23:54:48

just use a different variable:

its remainder two because the two cannot be divided by 4 to give an integer

Last edited by luca-deltodesco (2007-05-23 23:55:09)

JaneFairfax · 2007-05-23 23:58:23

The two odd numbers should be 2r+1 and 2s+1. (2r+1 and 2r−1 are merely two odd numbers with a difference of 2.)

Daniel123 · 2007-05-24 00:01:57

Im only 15.... what does mod mean?!

luca-deltodesco · 2007-05-24 00:07:58

basicly, mod is division, but with remainders.

so

basicly means that 4 divides the left hand expression, leaving 2 behind

Last edited by luca-deltodesco (2007-05-24 00:08:19)

Identity · 2007-05-24 00:09:28

mod is short for 'modulo' which means 'remainder'.

Means that a and b, when divided by n, both leave the same remainder.

e.g

Means that 13 and 5, when divided by 4, leave the same remainder.

You shouldn't be too concerned about the weird sign, it just means that 13's remainder, when divided by 4, is equivalent to 5's remainder, when divided by 4 (they both have remainders of 1), therefore the remainders are 'equivalent'. The sign means 'equivalence'.

Last edited by Identity (2007-05-24 00:18:30)

Daniel123 · 2007-05-24 00:10:51

oooohhh i get it now.. thank you! what are the chances of me learning that next year do you think? (AS level)

Last edited by Daniel123 (2007-05-24 00:12:36)

Sekky · 2007-05-24 01:51:39

Daniel123 wrote:

oooohhh i get it now.. thank you! what are the chances of me learning that next year do you think? (AS level)

Didn't you just learn it?

Don't rely on A-level teaching, read ahead, and your "chances" of learning something are certain, because you can do it yourself.

Identity · 2007-05-24 01:54:12

I heard from other here that in some places you learn it early, for instance in 10th grade, and in other places you have to wait till uni.

Daniel123 · 2007-05-24 01:55:33

lol i like your thinking! i clearly used the word 'learn' inappropriately... i meant 'be examined on it'! anyway i joined this forum today... and have never been on the computer for so long

Sekky · 2007-05-24 01:55:49

Identity wrote:

I heard from other here that in some places you learn it early, for instance in 10th grade, and in other places you have to wait till uni.

What are you talking about? You don't have to wait for anything, it's written right there!

Daniel123 · 2007-05-24 01:57:59

10th grade... what age is that? im from england... i dont know grades! im 15 though

Sekky · 2007-05-24 02:00:43

Daniel123 wrote:

lol i like your thinking! i clearly used the word 'learn' inappropriately... i meant 'be examined on it'! anyway i joined this forum today... and have never been on the computer for so long

Well welcome to the forum! If you're in your GCSEs at the minute then most likely the first thing you'll meet next year will be single variable calculus. Everyone I know went through a phase of first seeing calculus and screaming "Holy hell...this is real maths!", but a year later they look back at how trivial it seems by comparison to the absolute mammoth you've just awoken.

So if you want to start early for next year, calculus is probably the way to go.

luca-deltodesco · 2007-05-24 02:02:07

your doing your gcse's you said, so you're in the same year as me

10th grade in America is equivalent to our year10 in terms of years of study, but they dont leave after year11, they have to stay through to year13

they also start school a year later in america, so year11 age range is equivalent to grade10 age range

Last edited by luca-deltodesco (2007-05-24 02:02:34)

Daniel123 · 2007-05-24 02:02:09

Thanks for the advice... i dont think i will be taking on too much extra work until my GCSEs are out the way.... but in the summer i will be happy to do some especially with 2 and a half months off

luca-deltodesco · 2007-05-24 02:03:17

Daniel123 wrote:

especially with 2 and a half months off

the extended summer holiday is nice isnt it?

Sekky the jammy naughty has already finished his year at uni

Daniel123 · 2007-05-24 02:04:49

yes it is i cant wait.... but i think i may get slightly bored towards the end

Sekky · 2007-05-24 02:05:27

luca-deltodesco wrote:

Daniel123 wrote:
especially with 2 and a half months off
the extended summer holiday is nice isnt it?
Sekky the jammy naughty has already finished his year at uni

Oh yeah, three and a half months till the start of the year.

Lots of maths to do for then.

Math Is Fun Forum

#1 2007-05-23 23:48:13

Prove algebraically - odd numbers

#2 2007-05-23 23:54:48

Re: Prove algebraically - odd numbers

#3 2007-05-23 23:58:23

Re: Prove algebraically - odd numbers

#4 2007-05-24 00:01:57

Re: Prove algebraically - odd numbers

#5 2007-05-24 00:07:58

Re: Prove algebraically - odd numbers

#6 2007-05-24 00:09:28

Re: Prove algebraically - odd numbers

#7 2007-05-24 00:10:51

Re: Prove algebraically - odd numbers

#8 2007-05-24 01:51:39

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#9 2007-05-24 01:54:12

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#10 2007-05-24 01:55:33

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#11 2007-05-24 01:55:49

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#12 2007-05-24 01:57:59

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#13 2007-05-24 02:00:43

Re: Prove algebraically - odd numbers

#14 2007-05-24 02:02:07

Re: Prove algebraically - odd numbers

#15 2007-05-24 02:02:09

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#16 2007-05-24 02:03:17

Re: Prove algebraically - odd numbers

#17 2007-05-24 02:04:49

Re: Prove algebraically - odd numbers

#18 2007-05-24 02:05:27

Re: Prove algebraically - odd numbers

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