4. If p is a prime then p2
prime then p2 = 1 mod 24
Answers
Step-by-step explanation:
Since 504=2
3
.3
2
.7, therefore it is enough to show that p
6
−1 is a multiple of 2
3
,3
2
and7.
Step 1: Divisibility by 8:
p
6
−1 is divisible by p
2
−1. Since p is a prime greater than 7, therefore p−1 and p+1 are both even. Out of the consecutive even integers p−1 and p+1, one must be a multiple of 2 and the other must be a multiple of 4. (In fact, if p−1=4k+2, then p+1=4k+4;
if p−1=4k,thenp+1=4k+2. In either case (p−1)(p+1) is a multiple of 8).
Step 2: Divisibility by 9:
Since the product of three consecutive integers p−1,p,p+1 is divisible by 3,and p is not divisible by 3 (because it is a prime greater than 3), therefore (p−1)(p+1) is divisible by 3.
Now p
6
−1=(p
2
−1)(p
4
+p
2
+1),
=(p
2
−1){(p
2
−1)+3p
2
}
Since p
2
−1 is divisible by 3, therefore (p
2
−1)
2
+3p
2
is also divisible by 3. Consequently p
6
−1 is divisible by 9.
Step 3: Divisibility by 7:
Since 7 is prime 7 and p is prime to 7 (being a prime greater than 7), therefore by Fermat's theorem p
6
−1 is multiple of 7.
Since p
6
−1 is divisible by 8, 9 and 7 and the numbers 8, 9 and 7 are co-prime, therefore it is divisible by 8×9×7,i.e.,504