SOLUTION: Set A consists of all integers a such that a = p2 – 1, where p is a prime greater than 3. What is the greatest common factor of all the numbers in set A?

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Question 1033719: Set A consists of all integers a such that a = p2 – 1, where p is a prime greater than 3. What is the greatest common factor of all the numbers in set A?
Answer by robertb(5830) About Me  (Show Source):
You can put this solution on YOUR website!
a+=+p%5E2+-+1+=+%28p-1%29%28p%2B1%29 ==> either p-1 or p+1 is divisible by 3, since in any three consecutive natural numbers {p-1, p, p+1}, one of them is divisible by 3, and definitely it is not p because it is prime.
Also, p being a prime greater than or equal to 5 means it is an odd number and should be of the form p = 2k+1.
==> a+=+p%5E2+-+1+=+%28p-1%29%28p%2B1%29+=+2k%282k%2B2%29+=+4k%28k%2B1%29.
The expression 4k(k+1) has two factors that are consecutive numbers, hence one of them has 2 as a factor.
==> 8 divides a+=+p%5E2+-+1.
==> 3*8 = 24 is common factor of all numbers a. Since 24+=+5%5E2+-+1, it also makes it the greatest common factor.