Find all such Natural number n such that 7 divides 5^n + 1

Discussion:

Teacher: This is simple case of modular arithmetic. Consider the set of all residues of 5^n modulo 7 and you will see a pattern

Student: Sure. 5^1 = 5 , 5^2 = -3, 5^3 = -1 , 5^4 = 2 , 5^5 = 3 , 5^6 = 1 modulo 7. The residues repeat after that since 5^{6k + m} = 5^{6k} \times 5^m = 5^m modulo 7 where m \le 6 .

We want 5^n = -1 mod 7 hence n must be an odd multiple of 3 or n = 6r + 3 for any nonnegative integer r.