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#### Calculate the number of ways 2 numbers can be chosen from the set containing perfect square positive integers till 111 and multiplied together to obtain a product that is a multiple of 4 or 6.Option: 1 3Option: 2 8Option: 3 6Option: 4 4

To calculate the number of ways two numbers can be chosen from the set of perfect square positive integers up to 111 and multiplied together to obtain a product that is a multiple of 4 or 6, we need to consider the factors of 4 and 6.

Perfect square positive integers up to 111 are:

1, 4, 9, 16, 25, 36, 49, 64, 81, 100

To find the numbers that are multiples of 4 and 6, we need to find their common multiples.

The common multiples of 4 and 6 are: 12, 24, 36, 48, 60, 72, 84, 96, 108

Now, let's count the number of perfect square positive integers in this set that are multiples of 4 and 6.

Numbers that are perfect squares and multiples of 4 and 6: 36, 64, 100

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 perfect square positive integers up to 111 and multiplied together to obtain a product that is a multiple of 4 or 6 is (3 choose 2) = 3 ways.

Thus, there are 3 ways to choose two numbers from the set of perfect square positive integers up to 111 such that their product is a multiple of 4 or 6.