Question 3 a) Let a ∈ℤ^+ be a composite number. Then prove...

Question 3 a) Let $a \in \mathbb{Z}^+$ be a composite number. Then prove that there exists a prime number $p$ such that $p|a$ and $p \le \sqrt{a}$. b) Explain whether or not 577 is a prime number.

Transcript

00:01 Hello here we have objective to prove that in both p and 8p1 is minus 1 and prime and then 8p plus 1 must be composite.

00:10 Now for the first part we have to prove that we have a given that is p and 8p minus 1 are prime.

00:23 Now proof for 8p plus 1 is a composite use the fact that there cannot be have three consecutive numbers that are not divisible by three and from here we can get it as p and 8p minus 1 are being prime.

00:40 So from here we can say that p and 8p minus 1 are being prime.

00:54 So this is prime are not divisible by divisible by three.

01:07 So from here but we can say that that is 8p plus 1 being the third being the third term third term has to be divisible by three as to be divisible by three.

01:36 Now from here we can say even when p is equal to even when p is equal to three and 8p minus 1.

01:49 So here putting the value of p equal to three so we have 8 multiplied by p that is equal to 24 and 24 minus 1 that is gives us 23 and also for 8p plus 1.

02:01 Now putting p equal to three that is so 8 multiplied by 3 that is 24 plus 1 that is give 25.

02:08 So from here we can say that which is this is composite.

02:16 So from here hence finally from here we can say that that is 8p plus 1 is composite.

02:28 So this is the proof for the first part.

02:31 Now for the second part we have to prove that that is we have to prove that is if p and 8p square plus 1 are prime then 8p square minus 1 are is prime.

02:42 So now for the second part here we have to prove that is that it is given again that is p and 8p that is we have a given that is 8p square plus 1 are prime...

Question

Question 3 a) Let $a \in \mathbb{Z}^+$ be a composite number. Then prove that there exists a prime number $p$ such that $p|a$ and $p \le \sqrt{a}$. b) Explain whether or not 577 is a prime number.

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