Show that 4^(3p)+1 is divisible by 5

[KarlyD]

Show that if p is an odd number, [4^(3p)]+1 is divisible by 5. [It is actually divisible by 65.]

My work:
p=odd
4^(3p)+1=divisible by 65

let p=1
4^(3x1) + 1 = 64 + 1
=65

let p=3
4^(3x3) + 1 = 262144
=262145

let p=5
4^(3x5) + 1 = 1073741824
=1073741825

**I see that these are all divisible by 5 (or 65), but I don't know how to write a general statement for it, including all odd numbers.

The question is worth 6 marks and I have no idea how to get those 6 marks! Please help!

 

[stapel]

Have you studied induction proofs yet?

Note: To say that p is odd is to say that, for some natural number n (assuming we're sticking to non-negative values of p), you have p = 2n + 1. Then 4^3p + 1 = 4^{3(2n + 1)} + 1 = 4^{6n + 3} + 1 = 64 (4⁶ⁿ) + 1.

Eliz.

 

[KarlyD]

Thanks a lot!!!!!!
:lol: