Let a, b and c integers. Prove or disprove the next statements: a.7|𝑎^7−𝑎 ; b. If 7|𝑎2+𝑏2 then 7|𝑎 and 7|𝑏 ; c. If 7|𝑎3+𝑏3+𝑐3 then 7|𝑎𝑏𝑐? Please provide guidance.
To prove or disprove the given statements, we can use basic number properties and divisibility rules. Let's go through each statement one by one.
a. To prove or disprove whether 7 divides a^7 - a, we need to check if a^7 - a is divisible by 7 for all integers a.
To do this, you can evaluate a^7 - a for different values of a, and check if the result is divisible by 7. If you find a counterexample where the result is not divisible by 7, that would disprove the statement. However, if you observe that the result is always divisible by 7, then you can conclude that the statement holds true.
b. To prove or disprove if 7|a^2 + b^2 implies 7|a and 7|b, we can use the fact that if a number is divisible by 7, then its square is also divisible by 7. To prove this statement, you can consider two cases:
1. If a is divisible by 7, then a^2 is divisible by 7. In this case, a^2 + b^2 will also be divisible by 7.
2. If a is not divisible by 7, then it means a^3 is not congruent to 0 (mod 7). In this case, you can check all the possible residues of a^2 (0, 1, 2, 3, 4, 5, 6) modulo 7 and verify that a^2 + b^2 is not divisible by 7 for any non-zero residue of a^2.
c. To prove or disprove if 7|a^3 + b^3 + c^3 implies 7|abc, we can again use the congruence properties and consider different cases:
1. If any of a, b, or c is divisible by 7, then a^3 + b^3 + c^3 will be divisible by 7. In this case, you can conclude that 7|abc.
2. If none of a, b, and c are individually divisible by 7, you can check all the possible residues of a^3, b^3, and c^3 modulo 7 and verify that a^3 + b^3 + c^3 is not divisible by 7 for any combination of residues.
Remember, to disprove a statement, you only need to find one counterexample. On the other hand, to prove a statement, you need to show that it holds true for all possible values.
By following these guidelines and carrying out the necessary calculations, you can prove or disprove the given statements.