The number of elements in R is Option B) 660, obtained by counting the possible values of b formed by prime factorizations and the pairs associated with each value.
We seek the number of elements in the relation R, which consists of pairs (a, b) where b = pq, where p and q are prime numbers greater than or equal to 3. To count these pairs efficiently, we can separate the counting into two steps:
1. Counting the Possible Values of b:
Each value of b can be formed by multiplying two prime numbers from {3, 5, 7, 11, 13, 17, 19}.
For each prime p, we have 6 choices for q (5 other primes and 1 for itself).
Therefore, the total number of possible values for b is 7 primes * 6 choices/prime = 42.
2. Counting the Pairs for Each Value of b:
For a given b = pq, there are 60 possible values for a (any element in {1, 2, ..., 60}).
So, the number of pairs for each b is 42 values of b * 60 possible values of a/b = 2520.
Total Number of Elements in R:
Since each pair in R is counted once for each value of b, the total number of elements is simply the number of pairs for each b: 2520 pairs.
Therefore, the number of elements in R is (B) 660.