1. Write the reaction idea

Aniline reacts to give acetanilide in a $1:1$ mole ratio.

So, $n(\text{aniline}) = n(\text{acetanilide})$

2. Molar masses

• Aniline: $\mathrm{C_6H_5NH_2} = \mathrm{C_6H_7N}$ $M = 6\times 12 + 7\times 1 + 14 = 72 + 7 + 14 = 93\,\text{g mol}^{-1}$

• Acetanilide: $\mathrm{C_8H_9NO}$ $M = 8\times 12 + 9\times 1 + 14 + 16 = 96 + 9 + 14 + 16 = 135\,\text{g mol}^{-1}$

3. Moles of aniline taken

Given mass of aniline $= 1.86\,$g

$n(\text{aniline}) = \frac{1.86}{93} = 0.02\,\text{mol}$

Therefore, $n(\text{acetanilide formed}) = 0.02\,\text{mol}$

4. Theoretical mass of acetanilide

$m = nM = 0.02 \times 135 = 2.70\,\text{g}$

5. Loss during purification

$10\%$ of product is lost, so $90\%$ is obtained.

$m_{\text{obtained}} = 0.9 \times 2.70 = 2.43\,\text{g}$

6. Match with the required form

Given form: $\underline{\hspace{2cm}} \times 10^{-2}$

Now, $2.43 = 243 \times 10^{-2}$

So the required integer is: $\boxed{243}$