Given:
- Total equilibrium energy of the system = 138.06 KaT
- Three states with energies E = 0, KBT, and 2KBT
- N distinguishable particles are distributed among these states

To find:
- The number N of particles

Approach:
1. Let's assume that the number of particles in the states with energies 0, KBT, and 2KBT are x, y, and z respectively.
2. We know that the total energy of the system is given by the sum of the energies of all the particles in each state.
3. Using this information, we can write the equation:

0x + (KBT)y + (2KBT)z = 138.06 KaT ...(1)

4. Since the particles are distinguishable, the number of particles in each state can be any positive integer value.
5. We need to find the values of x, y, and z that satisfy equation (1) and are positive integers.

Solution:
1. Let's solve equation (1) to find the values of x, y, and z.
2. Since KBT is a common factor, we can divide equation (1) by KBT to simplify the equation:

0x + y + 2z = 138.06/KBT ...(2)

3. From equation (2), we can see that y and z must be positive integers to satisfy the given conditions.
4. Let's find the possible values of y and z that satisfy equation (2) and are positive integers.
5. We can start by assuming different values of z and find the corresponding value of y using equation (2).
- Let's assume z = 1. Substituting this value in equation (2), we get:
y = 138.06/KBT - 2 ...(3)
- Similarly, assuming z = 2, we get:
y = 138.06/KBT - 4 ...(4)
- We continue this process until we find a positive integer value for y.
6. To solve equation (3) or (4), we need the value of KBT.
7. Since the value of KBT is not given in the question, we cannot determine the exact values of y and z.
8. However, we can make an educated guess by assuming different values of KBT and checking if the resulting values of y and z are positive integers.
9. Let's assume KBT = 1 and substitute this value in equations (3) and (4).
- From equation (3), y = 138.06 - 2 = 136.06. Since y is not a positive integer, this assumption is incorrect.
- From equation (4), y = 138.06 - 4 = 134.06. Again, y is not a positive integer.
10. We can continue this process by assuming different values of KBT until we find the correct values of y and z that are positive integers.
11. Once we find the values of y and z, we can substitute them back into equation (1) to find the value of x.
12. The final step is to find the sum of x, y, and z, which will give us the number N of particles.