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Let S = 1, 2, 3, , N . A relation R on the set S is defined as R = (x, y) : x is a prime number and y is a perfect square , where x, y S . If the total number of elements in R is 105 , then the sum of all possible values of N is

Options

  1. A49
  2. B255
  3. C153
  4. D202

Correct answer

D. 202

Step-by-step solution

The number of elements in the relation R is given by the product of the number of valid choices for x and the number of valid choices for y . Let P(N) be the number of primes less than or equal to N , and S(N) be the number of perfect squares less than or equal to N . We are given that |R| = P(N) S(N) = 105 . The factors of 105 are (1, 105), (3, 35), (5, 21), (7, 15) . Since S(N) = N , we can test the possible values for S(N) and P(N) . Case 1: S(N) = 7 and P(N) = 15 . If S(N) = 7 , then 7 N Now we find the 15^ th

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