Question

the relation a^n|b^n implies that a|b. I know its true but i don't know how to go about writing it out in proof form

Answer 1

I suppose proof by induction.

Answer 2

\(a^n | b^n \implies b^n = a^nk\) for some \(k\)

\(\large \implies \frac{b^n}{a^n} - k = 0 \)

\(\large \implies \left(\frac{b}{a}\right)^n - k = 0 \)

Clearly b/a is a zero of above polynomial, that means \(\large a|b\) since the only possible rational zeroes of polynomial \(\large x^n-k = 0\) are integers

Answer 3

induction gives a nice proof too, try it :)

Answer 4

thank you

Answer 5

For induction :

base case : n=1 is true since a^1 | b^1 => a|b

assuming a^k|b^k => a|b, you need to prove a^k+1|b^k+1 => a|b:

\(\large a^{k+1}|b^{k+1} \implies aa^k |bb^k \implies a | bk \implies a|b \)

Answer 6

you need to justify each step a bit thoroughly if you're doing this proof as homework

Answer 7

fixed a typo in last line :

\(\large a^{k+1}|b^{k+1} \implies aa^k |bb^k \implies a | b\color{Red}{t} \implies a|b\)

Answer 8

thank you for the help! it makes much more sense now