# how to prove cubic root of 25 is irrational

how to prove cubic root of 25 is irrational using mathematical induction?

I've been trying to do it for hours but can't get it, help plz guys :S

-
Do you know why the square root of two is irrational? Can you modify the proof? –  Alexander Thumm Oct 12 '12 at 7:11
Why did you tag this with the Induction tag ? Do you know about rings ? –  Belgi Oct 12 '12 at 7:19
Proofs by induction are for questions like "Prove such and such for all $n$." Since you are concerned with the single number $25$, proof by induction is not appropriate here. –  Austin Mohr Oct 12 '12 at 7:21
What does plz mean? Is that a short for "Platz" in German? –  Asaf Karagila Oct 12 '12 at 7:21
@asaf But what would "Hilfe Platz Jungs: S" then mean? –  draks ... Oct 12 '12 at 7:29

Let's suppose $r$ is a rational such that $r^3 = 25$, then there exist $a, b\in \mathbb N$ coprimes such that $$a^3 = 25\, b^3 \,.$$ The above relation implies the following chain $$5 \mid a^3 \implies 5 \mid a \implies 5^3 \mid 25\, b^3 \implies 5 \mid b^3 \implies 5 \mid b$$ but that contradicts the assumption that $a$ anb $b$ are relatively prime.