The ‘Prime’ Importance of Prime Numbers

In the previous article we discussed a few Data Sufficiency examples on Prime numbers. Let’s  enjoy some more.

1. A Pierpont prime is a prime number ‘t’, such that t = 2^k X 3^m + 1 . Where, k and m are non-negative integers. If  x is an integer, is x a Pierpont prime?

I. 2 <= x <= 4
II. 1 <= x <=3

Solution:

We know, that x will be a Pierpont prime if x could be represented in the form (2k X 3m +1) . And k and m are non-negative integers, so the first Pierpont prime will be when k are lowest i.e. 0.

So the lowest possible Pierpont prime = (2⁰ X 3⁰ + 1) =2,

Likewise, we also get (2⁰ X 3⁰ + 1) = 3 as a Pierpont prime.

Next,  (2⁰ X 3⁰ + 1) = 4, but beware 4 is not a prime!

So next will be, (2¹ X 3¹ + 1) = 7.

Statement I: 2 <= x <= 4

So, x = 2, or 3 or 4.

If  is 2 or 3 it is a Pierpont prime, which gives us YES as the answer for the question, but if  x = 4, as 4 is not a prime, therefore not a Pierpont prime, so the answer is NO.

Thus, we don’t get specific answer for the question asked, therefore, answer options A and D are struck out.

Statement II: 1 <= x <=3

So, x =1, or 2 or 3.

Here again, 1 is not a prime, whereas 2, and 3 are Pierpont prime. So, x may be the Pierpont prime or not.

Thus, we don’t get specific answer for the question asked, therefore, answer options B is struck out.

Therefore, combining statements I and II:

x = 2 or 3.

In either case, x is a Pierpont prime, so we get a specific answer of YES to the question asked.

Answer: C

2. If is a non- negative integer having exactly two distinct positive factors, and , what is value of ?

I. There are exactly 59 prime numbers between 1 and (x + 1)
II. There are exactly prime numbers between 1 and 278.

Solution:  This is a very interesting way of GMAC testing you on the end objective!

We need to understand here that the bottom line is to see whether we will be able to find the value of x or not. We are not supposed to actually sit and calculate the possible value for x – this is a Data sufficiency question and not a problem solving.

x is a non- negative integer having exactly two distinct positive factors, means that x is a prime number.

Statement I : It is clear from the statement that between 1 and (x + 1) there are exactly 59 prime numbers, thus, x is definitely the 59^th prime number. As, there is only one pair of prime numbers at the distance of 1 unit – 2 and 3.

So, we got that x is 59^th prime number, can we find it, Yes! Are we supposed to find it, NO!!  (its enough for us to determine whether we are able to get one specific answer or not). Thus, we will be getting a specific answer. Lets strike out answer options, B, C and E.

Statement II : Ok, now will we be able to check how many prime numbers are there between 1 and 278, Yes. Are we supposed to find out that, No not really (Its already mentioned that x is a prime number). There will certainly be a specific number of prime numbers between any two given numbers, its not going to change later.

So, we will be able to get a specific answer using the information given in this statement. Thus, strike out A.

Answer: D.