#### Determine the number of ways two numbers from the prime number set upto 200 and multiplied together to obtain a product that is a multiple of neither 5 nor 3.Option: 1  420  Option: 2 861  Option: 3 816Option: 4 960

To determine the number of ways two numbers from the set of prime numbers up to 200 can be multiplied together to obtain a product that is not a multiple of 5 or 3, we need to exclude the prime numbers that are divisible by 5 or 3.

First, let's list the prime numbers up to 200:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199

Numbers divisible by 5: 5, 10, 15, 20, ..., 195, 200

Numbers divisible by 3: 3, 6, 9, 12, ..., 195, 198

To find the number of ways to choose two numbers such that their product is not a multiple of 5 or 3, we need to exclude the numbers divisible by 5 or 3 and consider the remaining prime numbers.

Excluding the prime numbers divisible by 5 or 3, we have the following set of prime numbers:

2, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199

Now, let's count the number of prime numbers in this set. There are 42 prime numbers.

To choose two numbers from this set, we can use the formula for combinations:
$\mathrm{n C r=n ! /(r !(n-r) !)}$

Therefore, the number of ways to choose two numbers from the set of prime numbers up to 200 and multiply them together to obtain a product that is not a multiple of 5 or 3 is (42 choose 2) = 861 ways.

Thus, there are 861 ways to choose two numbers from the prime number set up to 200 and multiply them together to obtain a product that is a multiple of neither 5 nor 3.