Pure Maths Help

Let a be an integer.
(a) Prove that a^3 ? k mod 7, where k is an element of {0, 1, 6}, i.e. the only cubic remainders modulo 7 are 0,1 and 6. [10]
(b) Show, using part (a) or otherwise that the equation 7(x^4) + 2(y^3) = 3 has no integer solutions. [10]
(c) Prove that 5 does not divide a^3 + a^2 + 1

Find explicit examples of
(a) an injective map f1 : N (arrow) Z ;
(b) a surjective map f2 : Z (arrow) N ;
(c) a surjective map f3 : N (arrow) Z ;
(d) an injective map f4 : Z (arrow) N .

Hi, I'm stuck on these questions and any help would be really helpful. For the second one i'm aware of what injective and surjective mean (well kinda) just unsure how you would find the examples as they switch from N to Z and stuff.

Thanks

(edited 13 years ago)

Reply 1

username22878

What I just posted complicated matters unnecessarily so I'll try again. For the second part, variations on just one function/idea will work - any ideas?

Reply 2

Wha'sHisFace

I was thinking maybe for b in the second question about the maps that perhaps you could do |x| but do |x| + 1 so that wen we put x as zero we get an answer of 1 which is in the N field. Is that anywhere near right?

Reply 3

username22878

Original post by Wha'sHisFace

That certainly works. Thing is, there's actually a bijection between N and Z (we say Z is 'countable'), so if you can figure out what that is, then the same map (or its inverse) will work in all four cases.

(If you have encountered countability already, then you should know what the bijection is. If not, then it's a fun thing to work out.)

EDIT: also, a bit of pedantry. N isn't a field.