Sure, let's solve the problem step-by-step.
### Step 1: Identify the problem
We need to find the smallest five-digit number that is divisible by three given numbers: 12, 15, and 20.
### Step 2: Calculate the Least Common Multiple (LCM)
To find a number that is divisible by 12, 15, and 20, we first need to calculate the Least Common Multiple (LCM) of these numbers.
#### Find the prime factorization:
- 12 = 2² × 3
- 15 = 3 × 5
- 20 = 2² × 5
#### Determine the highest powers of all prime factors:
- The highest power of 2 is 2² (from 12 or 20).
- The highest power of 3 is 3 (from both 12 and 15).
- The highest power of 5 is 5 (from both 15 and 20).
#### Calculate the LCM:
\( \text{LCM} = 2^2 \times 3 \times 5 \)
\( \text{LCM} = 4 \times 3 \times 5 \)
\( \text{LCM} = 12 \times 5 \)
\( \text{LCM} = 60 \)
So, the LCM of 12, 15, and 20 is 60.
### Step 3: Determine the smallest five-digit number
The smallest five-digit number is 10000. We need to find the smallest five-digit number that is divisible by 60.
### Step 4: Find the smallest five-digit number divisible by 60
To find this, we can divide 10000 by 60 and take the ceiling of the quotient, then multiply it back by 60.
1. Divide 10000 by 60:
[tex]\[ 10000 \div 60 = 166.6667 \][/tex]
2. Take the ceiling of the quotient:
[tex]\[ \text{Ceiling}(166.6667) = 167 \][/tex]
3. Multiply the ceiling value back by 60:
[tex]\[ 167 \times 60 = 10020 \][/tex]
### Final Answer
The smallest five-digit number that is divisible by 12, 15, and 20 is 10020.
