Question

р-простое число р>3 а)докажите что, р*р-1 деоится на 12б)р=6к+-1, к принадлежит N

Answer 1

докажем сначала пункт б)

каждое натуральное число можна записать в виде 6k+1,6k+2, 6k+3, 6k+4, 6k+5, (то же самое что 6l-1), 6k+6, где k=0, или k - натуральное (так как при делении на 6 остатки могут быть 0,1,2,3,4,5)

числа вида 6k+2, 6k+4, 6k+6 четные поэтому делятся на 2, но но одно простое число больше 3 на 2 не делится, поэтому среди чисел этого вида нет простых

числа вида 6k+3=3*(2k+1) делятся на 3, но ни одно число большее 3, на 3 не делится, поэтому среди чисел данного вида нет протых чисел, поэтому простые числа находятся срди чисел вида р=6к+-1, к принадлежит N, что и требовалось доказать

 

теперь используя доказанный пункт б) докажем а)

р*р-1=(p-1)(p+1)  - по формуле разности квадратов

рассмотрим два возможных случая

первый р=6k+1, к принадлежит N

тогда

р*р-1=(6k+1-1)(6k+1+1)=6k*(6k+2)=12k*(3k+1), а значит деится на 12

второй p=6k-1

p*p-1=(6k-1-1)(6k-1+1)=(6k-2)*6к=12к*(3к-1), а значит делится на 12.

Доказано

 

Answer 2

Proof that p * (p - 1) is divisible by 12, where p is a prime number greater than 3

To prove that p * (p - 1) is divisible by 12, we need to consider two cases:

Case 1: p is divisible by 2 If p is divisible by 2, then p can be written as p = 2k, where k is an integer. Substituting this into the expression p * (p - 1), we get: p * (p - 1) = (2k) * (2k - 1) = 4k(k - 1)

Since k and (k - 1) are consecutive integers, one of them must be even and the other odd. Therefore, the product 4k(k - 1) is divisible by 8.

Case 2: p is not divisible by 2 If p is not divisible by 2, then p must be an odd prime number. In this case, p - 1 is an even number, and we can write p - 1 as p - 1 = 2k, where k is an integer. Substituting this into the expression p * (p - 1), we get: p * (p - 1) = p * 2k = 2pk

Since p is an odd prime number, it is not divisible by 2. Therefore, the product 2pk is divisible by 2 but not by 4.

In both cases, we have shown that p * (p - 1) is divisible by 2. Now, we need to show that it is also divisible by 3.

Proof that p * (p - 1) is divisible by 3: To prove that p * (p - 1) is divisible by 3, we need to consider three cases:

Case 1: p is divisible by 3 If p is divisible by 3, then p can be written as p = 3k, where k is an integer. Substituting this into the expression p * (p - 1), we get: p * (p - 1) = (3k) * (3k - 1) = 9k(k - 1)

Since k and (k - 1) are consecutive integers, one of them must be divisible by 2. Therefore, the product 9k(k - 1) is divisible by 6.

Case 2: p ≡ 1 (mod 3) If p ≡ 1 (mod 3), then p - 1 is divisible by 3. Substituting this into the expression p * (p - 1), we get: p * (p - 1) = p * 3k = 3pk

Since p is divisible by 3, the product 3pk is divisible by 3.

Case 3: p ≡ 2 (mod 3) If p ≡ 2 (mod 3), then p is not divisible by 3, but p - 1 is divisible by 3. Substituting this into the expression p * (p - 1), we get: p * (p - 1) = p * 3k = 3pk

Since p is not divisible by 3, the product 3pk is divisible by 3.

In all three cases, we have shown that p * (p - 1) is divisible by 3.

Therefore, combining the results from both divisibility tests, we can conclude that p * (p - 1) is divisible by both 2 and 3, which means it is divisible by 12.