Quesition: Prove (5)1/3 is irrational.
Solution: Assume (5)^1/3 is rational. There are m, n ∈ N s.t. m, n have no common factors other than 1 and (5)^1/3 = m/n. Then 5 = (m/n)^3, and so 5n^3 = m^3. Hence 5|m^3, and then 5|m: this is because: if m≡1 (mod5) then m^3 ≡1^3 ≡1 (mod5) so5?|m^3; if m≡2 (mod 5) then m^3 ≡2^3 ≡3 (mod5), so again 5?|m3; since we know 5|m^3,now we must have m≡0(mod5),i.e. 5|m. So m=5k for some k∈N. It follows: 5n^3 =(5k)^3 =5^3k^3,and then n^3 = 5^2k^3. Thus, 5|n^3. So 5|n. Therefore, 5 is a common factor of m and n, which contradicts the choice of m, n. We now conclude (5)^1/3 is irrational
1)What does m^3 ≡2^3 ≡3 (mod5) means? Where does 3 come from?
2)When should I use mod to prove a statement?