Divisibility Proof: 9^n-5^n is Divisible by 4

[odolwa99]

Can anyone help me confirm if I have solved this correctly?

Many thanks.

Homework Statement

Q. 9^n-5^n is divisible by 4, for n\in\mathbb{N}_0

The Attempt at a Solution

Step 1: For n=1...
9^1-5^1=4, which can be divided by 4.
Therefore, n=1 is true...


Step 2: For n=k...
Assume 9^k-5^k=4\mathbb{Z}, where \mathbb{Z} is an integer...1
Show that n=k+1 is true...
i.e. 9^{k+1}-5^{k+1} can be divided by 4
9^{k+1}-5^{k+1} => 9^{k+1}-9\cdot5^k+9\cdot5^k-5^{k+1} => 9(9^k-5^k)+5^k(9-5) => 9(4\mathbb{Z})+5^k(4)...from 1 above => 36\mathbb{Z}+5^k\cdot4 => 4(9\mathbb{Z}+5^k)

Thus, assuming n=k, we can say n=k+1 is true & true for n=2,3,... & all n\in\mathbb{N}_0

 

[micromass]

That's good. Some remarks on notation:

Assume 9^k-5^k=4\mathbb{Z}, where \mathbb{Z} is an integer...

Using \mathbb{Z} is not good here. That is the symbol for the set of all integers. You should write 9^k-5^k=4z where z is an integer.

9^{k+1}-5^{k+1} => 9^{k+1}-9\cdot5^k+9\cdot5^k-5^{k+1} => 9(9^k-5^k)+5^k(9-5) => 9(4\mathbb{Z})+5^k(4)...from 1 above => 36\mathbb{Z}+5^k\cdot4 => 4(9\mathbb{Z}+5^k)

The => should be =

 

[odolwa99]

Ok. Thank you.