To find the least 5-digit number that is exactly divisible by 20, 25, and 30, we need to follow these steps:
Step 1: Find the Least Common Multiple (LCM)
To determine the LCM of the numbers 20, 25, and 30, we first find the prime factorization of each number:
- 20 = 2² × 5¹
- 25 = 5²
- 30 = 2¹ × 3¹ × 5¹
Now, we take the highest power of each prime number:
- For 2: Highest power is 2² (from 20)
- For 3: Highest power is 3¹ (from 30)
- For 5: Highest power is 5² (from 25)
Now, we can calculate the LCM:
LCM Calculation:
- LCM = 2² × 3¹ × 5²
- LCM = 4 × 3 × 25
- LCM = 12 × 25 = 300
Step 2: Find the Least 5-Digit Number
The next step is to find the least 5-digit number, which is 10000. We need to find the smallest multiple of 300 that is greater than or equal to 10000.
Calculation:
To find this, we divide 10000 by 300:
- 10000 ÷ 300 = 33.33
We round this up to the nearest whole number, which is 34.
Step 3: Calculate the Multiple
Now, we multiply 300 by 34:
- 300 × 34 = 10200
Final Answer:
Thus, the least 5-digit number that is exactly divisible by 20, 25, and 30 is:
10200