The 2013 STEP Prep Thread!

Can someone explain to me why

n^{3}=0,1,6 (mod7)

? Looking at the residues mod 7 there's obviously a clear pattern, but I don't have an intuitive understanding.

How can you prove this rigorously without using induction on

n(n^{3}+1)(n^{3}-1)

?

Reply 782

Original post by und

Can someone explain to me why

n^{3}=0,1,6 (mod7)

? Looking at the residues mod 7 there's obviously a clear pattern, but I don't have an intuitive understanding.

How can you prove this rigorously without using induction on

n(n^{3}+1)(n^{3}-1)

?

if a^m= b mod n, then (nk+a)^m = b mod n. all you have to do is consider the residues of 0-6^3 mod 7 :wink:

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Reply 783

Dirac Spinor

10

Original post by und

Can someone explain to me why

n^{3}=0,1,6 (mod7)

? Looking at the residues mod 7 there's obviously a clear pattern, but I don't have an intuitive understanding.

How can you prove this rigorously without using induction on

n(n^{3}+1)(n^{3}-1)

?

argue by cases
0^3=0, 1^3=1, 2^3=8=7+1, 3^3=27=21+6, 4^3=64=63+1, 5^3=125=119+6, 6^3=216=210+6
EDIT:beaten by tmm

(edited 11 years ago)

Reply 784

Original post by TheMagicMan

if a^m= b mod n, then (nk+a)^m = b mod n. all you have to do is consider the residues of 0-6^3 mod 7 :wink:

This was posted from The Student Room's iPhone/iPad App

I can see that pattern but how do you go about proving the result you've just posted?

Reply 785

Original post by ben-smith

argue by cases
0^3=0, 1^3=1, 2^3=8=7+1, 3^3=27=21+6, 4^3=64=63+1, 5^3=125=119+6, 6^3=216=210+6
EDIT:beaten by tmm

Surely that's not rigorous? It's easy enough to find the pattern and state it, but how do you prove it?

Reply 786

Original post by und

I can see that pattern but how do you go about proving the result you've just posted?

Binomial theorem?

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Reply 787

Dirac Spinor

10

Original post by und

Surely that's not rigorous? It's easy enough to find the pattern and state it, but how do you prove it?

Those are the only possibilities...

a \equiv b mod(n) \Rightarrow a^r \equiv b^r mod(n)

Reply 788

Original post by TheMagicMan

Binomial theorem?

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I had a feeling that would be the case...

Original post by ben-smith

Those are the only possibilities...

a \equiv b mod(n) \Rightarrow a^r \equiv b^r mod(n)

I'm not familiar with the rules of modular arithmetic but isn't this the same as what TheMagicMan posted, just in different notation?

Reply 789

Dirac Spinor

10

Original post by und

I had a feeling that would be the case...

I'm not familiar with the rules of modular arithmetic but isn't this the same as what TheMagicMan posted, just in different notation?

I'd like to think I came up with it on my own :tongue:

Reply 790

Original post by und

I had a feeling that would be the case...

I'm not familiar with the rules of modular arithmetic but isn't this the same as what TheMagicMan posted, just in different notation?

Yes they're equivalent

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Reply 791

ian.slater

12

Original post by und

Can someone explain to me why

n^{3}=0,1,6 (mod7)

? Looking at the residues mod 7 there's obviously a clear pattern, but I don't have an intuitive understanding.

How can you prove this rigorously without using induction on

n(n^{3}+1)(n^{3}-1)

?

As others have said, you have to consider all the cases. This is called 'proof by exhaustion' and is quite valid provided you show that you have dealt with them all.

If you are looking for a 'deeper' explanation of 'why?', capable of a more powerful proof then that's above my pay grade. Last I heard the Four-colour theorem has only been proved by an inelegant proof by exhaustion, which leaves open the question of 'why?'.

Reply 792

shamika

I got a response from Imperial regarding their requirements:

Their application process works in two stages. In the first stage, they decide whether to give you an offer or not. As they no longer interview, this is based on the enthusiasm you show for the subject. Good examples would be:

- doing maths competitions (e.g. the Olympiad - if you get a gold in the first round say so and definitely trumpet getting into BMO 1 and above)

- seeking out and attending 'public lectures' or attending Summer Schools - check out the ones organised by the Sutton Trust, they sound spectacular. These are typically only available for those in non-privileged backgrounds.

- reading "serious books". Please do not make stuff up - no one will believe you if you say you've mastered an undergrad book (say Baby Rudin - look it up) if your A-Level grades aren't spectacular and you've should have also done other things on this list

- "learning more maths by yourself." For example, I learned about modular arithmetic reading What is Mathematics? and got as far as quadratic residues. That's the sort of thing that'll make anyone take notice (and to be fair was way over the top - I got a bit carried away when I realised how awesome pure maths was compared to A-Levels). Or you could read The Road to Reality and say something sensible about the sophistication of the underlying mathematical models used to describe physical phenomena

(the examples above come directly from Skoro, including the quotes)

He makes the point that one of the easiest ways to demonstrate commitment is to highlight you are planning on taking STEP. I therefore infer that many of you will mention it in your personal statements anyway. Mentioning STEP makes it more likely you'll get an offer but it will not by itself guarantee a place for you.

Note that Imperial still treat each individual application separately - including reading references and finding out a bit about the sort of school you currently attend (e.g. is it a state school or an elite independent). Therefore in isolation they may make offers which seem random but they have to look at the overall quality of students applying (see below for more on this)

Once they've decided to give you an offer, they then decide on the exact form. Again, there is discretion that can be applied - offers can and will be tweaked depending on circumstances.

- Typically they will give STEP offers where possible (and in fairness, a 2 in II or III is not an overly high barrier to reach)

- Where they do not give STEP offers, they will try to make offers based on 4 A2's (with A*A* in maths and further maths, obviously as a minimum)

- In "rarer cases" you may get an offer on just 3 A2's but you'll pretty much have to be exceptional. There is flexibility to take into account the school you've gone to (e.g. if its a state school where typically only 3 A2's is standard). This is not an excuse or a pity offer - you'll have to be pretty outstanding. The example given was a medal at IMO but that's obviously a bit extreme (if you're that outstanding getting an offer from Imperial should be the least of your worries)

So there you have it. I've asked a follow up to clarify the use of AEA rather than STEP, but let's give him some time to get back to me. He was great with agreeing to share all the above (even if none of it is earth-shattering) so I've already thanked him on behalf of all of you :smile:

I'll post a link to this in the maths applicants page too, in case people find it useful there too.

(edited 11 years ago)

Reply 793

Here is a rather nice problem for you maths junkies out there....

If I break a straight stick into n pieces, what is the probability that I can form an n sided polygon from the pieces?

Answer:

Spoiler

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Reply 794

Intriguing Alias

Original post by TheMagicMan

Here is a rather nice problem for you maths junkies out there....

If I break a straight stick into n pieces, what is the probability that I can form an n sided polygon from the pieces?

Answer:

Spoiler

This was posted from The Student Room's iPhone/iPad App

So there is a probability of 1/4 that you can make a 2 sides polygon?

You're lying.. :tongue:

Spoiler

Reply 795

Original post by Intriguing Alias

So there is a probability of 1/4 that you can make a 2 sides polygon?

You're lying.. :tongue:

Spoiler

Poisson distribution? :lol:

Reply 796

Original post by Intriguing Alias

So there is a probability of 1/4 that you can make a 2 sides polygon?

You're lying.. :tongue:

Spoiler

Well yes basically...this is a question that with a bit of hinting could easily appear on a step paper I think (it is probably too hard without hints (yes that's a challenge to you all :tongue:

))

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Reply 797

Original post by TheMagicMan

Well yes basically...this is a question that with a bit of hinting could easily appear on a step paper I think (it is probably too hard without hints (yes that's a challenge to you all :tongue:

))

This was posted from The Student Room's iPhone/iPad App

Spoiler

Never mind. I see where I went wrong with my assumption.

(edited 11 years ago)

Reply 798