**Calculating the molar mass of the insecticide:**

To determine the total number of moles of atoms in a 3.0mg sample of the insecticide, we first need to calculate the molar mass of the compound.

The molecular formula of the insecticide is C16H15Cl3O.

The molar mass of each element can be calculated by multiplying the atomic weight of the element by the number of atoms present in the formula.

The atomic weight of carbon (C) is 12.01, and there are 16 carbon atoms in the formula, so the total molar mass contribution from carbon is:

12.01 * 16 = 192.16 g/mol

The atomic weight of hydrogen (H) is 1.008, and there are 15 hydrogen atoms in the formula, so the total molar mass contribution from hydrogen is:

1.008 * 15 = 15.12 g/mol

The atomic weight of chlorine (Cl) is 35.45, and there are 3 chlorine atoms in the formula, so the total molar mass contribution from chlorine is:

35.45 * 3 = 106.35 g/mol

The atomic weight of oxygen (O) is 16.00, and there is 1 oxygen atom in the formula, so the total molar mass contribution from oxygen is:

16.00 * 1 = 16.00 g/mol

Adding up all the molar mass contributions:

192.16 + 15.12 + 106.35 + 16.00 = 329.63 g/mol

Therefore, the molar mass of the insecticide is 329.63 g/mol.

**Calculating the number of moles:**

Now that we know the molar mass of the insecticide, we can calculate the number of moles in a 3.0mg sample.

The number of moles can be calculated using the formula:

moles = mass / molar mass

In this case, the mass is 3.0mg and the molar mass is 329.63 g/mol. However, we need to convert the mass to grams before we can proceed with the calculation.

1mg = 0.001g

Therefore, the mass of the sample in grams is:

3.0mg * 0.001g/mg = 0.003g

Now we can calculate the number of moles:

moles = 0.003g / 329.63 g/mol ≈ 9.11 x 10^-6 moles

Therefore, the total number of moles of atoms in the 3.0mg sample of the insecticide is approximately 9.11 x 10^-6 moles.