Another Mathematical Proofs Problem?

Question

Prove 24 I (5^2n - 1)

{for those not familiar with this equation format, it means 24 is divisable into (5^2n - 1)}

i know that a I b could also be interpreted as ax = b, but this question is supposed to be done as an induction problem.

Answer 1

You can prove it by induction.

It is obviously true when n = 1.

Induction pypothesis: Suppose it is true for n

We need to show is true for (n+1).

Does 24 divide (5^(2*(n+1))-1))?

We can rewrrite this as:

25*(5^2n)-1 Subtract 24 from this to get 25*(5^2n)-25

This is just equal to 25*(5^2n)-1).

This has to be divisible by 24 -- since one of the factors is (5^2n)-1) and our induction hypothesis says that it is.

Add 24 to it and we get something that is also divisible by 24 -- but it is (5^(2*(n+1))-1)) -- so our induction is done.

Answer 2

If f isn't injective then for some a<>b in X, f(a)=f(b). yet then g(f(a))=g(f(b)). This although ability that a=b (because gof is injective), it really is a contradiction. wish this helps.