Here's how to find the common difference: Step 1: Write down the formula for the $n$-th term of an arithmetic progression (A.P.). The $n$-th term is given by $a_n = a + (n-1)d$, where $a$ is the first term and $d$ is the common difference. Step 2: Use the given information to form equations. "The sum of the 2nd and 5th terms of an arithmetic progression (A.P) is 42." The 2nd term is $a_2 = a + (2-1)d = a + d$. The 5th term is $a_5 = a + (5-1)d = a + 4d$. So, $a_2 + a_5 = 42$ $(a + d) + (a + 4d) = 42$ $$2a + 5d = 42 \quad \text{(Equation 1)}$$ "The difference between the 6th and 3rd terms is 12." The 6th term is $a_6 = a + (6-1)d = a + 5d$. The 3rd term is $a_3 = a + (3-1)d = a + 2d$. So, $a_6 - a_3 = 12$ $(a + 5d) - (a + 2d) = 12$ $a + 5d - a - 2d = 12$ $$3d = 12 \quad \text{(Equation 2)}$$ Step 3: Solve Equation 2 for $d$. $$3d = 12$$ $$d = \frac{12}{3}$$ $$d = 4$$ The common difference is 4. The final answer is $\boxed{\text{4}}$