When $a$ particular positive number is divided by 5, the remainder is 2. If the same number is divided by 6, the remainder is 1. If the difference between the quotients of division is 3, then find the number.

Step 1 (Formulate Equations): Let the positive number be $N$. Let the quotient when $N$ is divided by 5 be $q_1$, and when divided by 6 be $q_2$:

• $N=5q_1+2\text{— (Equation 1)}$

• $N=6q_2+1\text{— (Equation 2)}$

Step 2 (Use Quotient Difference): We are given that the difference between the quotients is 3. Since dividing by a smaller number (5) yields a larger quotient:

Step 3 (Equate the Equations and Solve): Substituting $q_1$ into Equation 1 and equating it to Equation 2:

Step 4 (Compute N): Using $q_2=16$ in Equation 2:

Step 5 (Conclusion): The number is exactly 97.