Question

12) Vaibhav wrote a certain number of positive prime
numbers on a piece of paper. Vikram wrote down
the product of all possible triplets among those
numbers. For every pair of numbers written by
Vikram, Vishal wrote down the coresponding
GCD. If 90 of the numbers written by Vishal were
prime, how many numbers did Vaibhav write?
(2011)
(a) 6
(b) 8
(c) 10
(d) Cannot be determined​

Answer 1

Step-by-step explanation:

Given Vaibhav wrote a certain number of positive prime  numbers on a piece of paper. Vikram wrote down  the product of all possible triplets among those  numbers. For every pair of numbers written by  Vikram, Vishal wrote down the corresponding  GCD. If 90 of the numbers written by Vishal were  prime, how many numbers did Vaibhav write?

Let there be n C 3 triplets. So Vikram wrote all triplets.

So for every pair of the products, Vishal wrote corresponding GCD in which there are 90 prime numbers.

1. Now GCD = 1 is not a prime and no number in common . (a,b,c) and (d,e,f)

2. Here GCD = a and a prime since one number is common. (a,b,c) and (a,d,e)

3. GCD = a x b and not prime since two numbers in common. (a,b,c) and (a,b,d)

4. GCD = a x b x c and not prime since three numbers in common. (a,b,c) and (a,b,c)            

Now there are 90 pairs where one number is common.  

Let n numbers be selected in n C1 = n ways

So next number be selected in (n – 1) C2 ways.

Also next number can be selected in (n – 3) C2 ways.

So total pairs will be n (n – 1) C2 x (n – 3) C2 / 2! (2! In order to avoid counting again)

So   n (n – 1) C2 x (n – 3) C2 / 2! = 90

      n (n – 1) C2 x (n – 3) C2  = 180

  So (n – 1) (n – 2) / 2! X (n – 3)(n – 4) / 2! = 180

 So n(n – 1)(n – 2)(n – 3)(n – 4) = 720

So n= 6 since 6 x 5 x 4 x 3 x 2 = 720  

So n = 6