Question

if p, p+2, p+4 are prime numbers then p= ___

(a) 5
(b) 3
(c) 29
(d) 191​

Answer 1

Answer:

Of the numbers p,p+2,p+4 one of the three is always divisibe by 3. That is easy to see: pmod3 is either 0, 1 or 2. If it is 0, then p is divisible by 3. If it is 1, then p+2 is divisible by 3. And if it is 2, then p+4 is divisible by 3.

So there is always a number that is divisible by 3. That number is not prime, unless that number happens to be 3, the only number that is prime and divisible by three. So the only solution is that p=3 , and indeed 3, 5, 7 are all primes.

Answer 2

Given,

p, p+2 and p+4 are prime numbers.

To find,

To check which of the given options (ie. 5,3,29,191) is correct value for p.

Solution,

We can simply solve this mathematical problem by using the following mathematical process.

Now,

According to the definition of a prime number, we come to know that : a prime number is only divisible by 1 and itself.

Here, we will put the given options in the place of p, and then we will inspect that p,p+2,p+4 are producing prime numbers or not. (inspection will be done according to the previously mentioned property of prime numbers.)

Now,

• If, p = 5, then p is a prime number because 5 is only divisible by 1 and 5
• If, p = 5, then p+2 = 7 is a prime number because 7 is only divisible by 1 and 7
• If, p = 5, then p+4 = 9 is not a prime number because 9 is divisible by 1,9 and 3 (factor other than 1 and number itself, arrives)

• If, p = 3, then p is a prime number because 3 is only divisible by 1 and 3
• If, p = 3, then p+2 = 5 is a prime number because 5 is only divisible by 1 and 5
• If, p = 3, then 3+4 = 7 is a prime number because 7 is only divisible by 1 and 7

• If, p = 29, then p is a prime number because 29 is only divisible by 1 and 29
• If, p = 29, then p+2 = 31 is a prime number because 31 is only divisible by 1 and 31
• If, p = 29, then p+4 = 33 is not a prime number because 9 is divisible by 1,33,3,11 (factors other than 1 and number itself, arrive)

• If, p = 191, then p is a prime number because 191 is only divisible by 1 and 191
• If, p = 191 , then p+2 = 193 is a prime number because 193 is only divisible by 1 and 193
• If, p = 191, then p+4 = 195 is not a prime number because 195 is divisible by 1,195,3and 5 (factors other than 1 and number itself, arrive)

• So, if p = 3, then only p,p+2 and p+4 become prime numbers.

Hence, value of p is 3