To answer your question, let's break it down into two parts:
a) Finding the number of primitive elements in the multiplicative group ℤ∗137:
To find the number of primitive elements in this group, we need to find the elements that generate the entire group when raised to different powers. In other words, we're looking for elements that have the maximum order, which is φ(137-1), where φ denotes Euler's totient function. For ℤ∗137, the order of any element must divide φ(137-1). Therefore, the number of primitive elements in this group is equal to φ(φ(137-1)).
b) Calculating the probability of randomly choosing a primitive element:
To find the probability, we need to divide the number of primitive elements by the total number of elements in the group, which is φ(137-1).
In conclusion, to determine the number of primitive elements in the multiplicative group ℤ∗137, you would need to calculate φ(φ(137-1)). The probability of randomly choosing a primitive element can be found by dividing the number of primitive elements by φ(137-1).