To determine the smallest capacity of a tank that can be filled exactly using two buckets, one with a capacity of 10 liters and the other with a capacity of 15 liters, we need to find the least common multiple (LCM) of the capacities of the two buckets.
Here is a step-by-step breakdown:
1. Identify the capacities of the buckets:
- Bucket 1 capacity: 10 liters
- Bucket 2 capacity: 15 liters
2. Find the prime factorizations of the capacities:
- Prime factorization of 10: [tex]\( 10 = 2 \times 5 \)[/tex]
- Prime factorization of 15: [tex]\( 15 = 3 \times 5 \)[/tex]
3. Determine the LCM:
- List all prime factors found in the factorizations: [tex]\(2, 3, 5\)[/tex].
- For each prime factor, use the highest power of that factor appearing in the factorizations.
- For 2: The highest power is [tex]\(2^1\)[/tex].
- For 3: The highest power is [tex]\(3^1\)[/tex].
- For 5: The highest power is [tex]\(5^1\)[/tex].
4. Calculate the LCM:
- The LCM is obtained by multiplying the highest powers of all prime factors:
[tex]\[
LCM = 2^1 \times 3^1 \times 5^1 = 2 \times 3 \times 5 = 30
\][/tex]
Thus, the least capacity of the tank that can be filled exactly using either the 10 liter or the 15 liter bucket is 30 liters.
