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
To prove or disprove the given statements, we need to apply modular arithmetic concepts. In modular arithmetic, we work with remainders when a number is divided by another number.
a. To prove/disprove 7|𝑎^7−𝑎,
We can use Fermat's Little Theorem. Fermat's Little Theorem states that if p is a prime number and a is an integer not divisible by p, then a^(p-1) ≡ 1 (mod p).
In this case, p = 7. Therefore, for any integer a not divisible by 7, we have a^6 ≡ 1 (mod 7).
Now, let's consider the specific case of a^7 − a.
If 7|a, then both a^7 and a are divisible by 7, so the statement is true.
If a is not divisible by 7, using Fermat's Little Theorem, we have a^6 ≡ 1 (mod 7). Therefore, a^7 ≡ a (mod 7) since a^6 ≡ 1 (mod 7). So, a^7 − a is divisible by 7.
Hence, the statement a. 7|𝑎^7−𝑎 is always true for any integer a.
b. To prove/disprove If 7|𝑎^2+𝑏^2 then 7|𝑎 and 7|𝑏,
Let's consider a counterexample to disprove the statement.
Let a = 1 and b = 3. Now, a^2 + b^2 = 1^2 + 3^2 = 10. As 7 does not divide 10, the statement b is false.
Therefore, the statement If 7|𝑎^2+𝑏^2 then 7|𝑎 and 7|𝑏 is false.
c. To prove/disprove If 7|𝑎^3+𝑏^3+𝑐^3 then 7|𝑎𝑏𝑐,
Let's consider a counterexample to disprove the statement.
Let a = 1, b = 2, and c = 3. Now, a^3 + b^3 + c^3 = 1^3 + 2^3 + 3^3 = 1 + 8 + 27 = 36. As 7 does not divide 36, the statement c is false.
Therefore, the statement If 7|𝑎^3+𝑏^3+𝑐^3 then 7|𝑎𝑏𝑐 is false.
So, the statements b and c are both false.