To solve this problem, we need to find the greatest common divisor (GCD) of the quantities of oil (403, 434, and 465 liters). The GCD will give the largest size of the barrels that can evenly divide all three quantities without mixing the oils.

Step 1: Find the GCD of 403, 434, and 465

1. Find the prime factorization of each number:

   • 403: $403 = 13 \times 31$ (prime factors).
   • 434: $434 = 2 \times 7 \times 31$ (prime factors).
   • 465: $465 = 3 \times 5 \times 31$ (prime factors).

2. Identify the common factors:

   • The common factor between $403, 434,$ and $465$ is $31$.

Step 2: The GCD is $31$

Thus, the largest size of each barrel is $31$ liters.

Step 3: Calculate the total number of barrels

• For $403$ liters: $\frac{403}{31} = 13$ barrels.
• For $434$ liters: $\frac{434}{31} = 14$ barrels.
• For $465$ liters: $\frac{465}{31} = 15$ barrels.

Final Answer:

The least possible number of barrels required is: $13 + 14 + 15 = 42 \text{ barrels}.$

Would you like me to explain the steps in more detail or perform a similar calculation?

Related Questions:

1. How is the GCD calculated using the Euclidean algorithm?
2. What if the oil quantities were in milliliters instead of liters?
3. How would the solution change if the barrels were of different sizes?
4. Can we automate the process of finding the GCD for any set of numbers?
5. What happens if we add a fourth type of oil with a specific quantity?

Tip:

When solving problems involving GCD, always double-check the prime factorization to avoid errors