Understanding the Problem
To find the least 5-digit number that is exactly divisible by 20, 25, and 30, we first need to understand what it means for a number to be divisible by these values. A number is divisible if it can be divided by another number without leaving a remainder.
Finding the Least Common Multiple (LCM)
To solve this, we need to find the LCM of 20, 25, and 30. The LCM is the smallest number that all the original numbers can divide into evenly.
- Prime Factorization:
- 20 = 2² × 5¹
- 25 = 5²
- 30 = 2¹ × 3¹ × 5¹
- Take the highest powers of all prime factors:
- 2: 2² (from 20)
- 3: 3¹ (from 30)
- 5: 5² (from 25)
- Calculate the LCM:
- LCM = 2² × 3¹ × 5² = 4 × 3 × 25 = 300
Finding the Least 5-Digit Number
Now that we have the LCM, which is 300, we need to find the smallest 5-digit number that is a multiple of 300.
- The smallest 5-digit number is 10000.
- Divide 10000 by 300:
- 10000 ÷ 300 = 33.33 (approximately)
- Round up to the next whole number:
- The next whole number is 34.
- Multiply by 300:
- 34 × 300 = 10200
Conclusion
The least 5-digit number that is exactly divisible by 20, 25, and 30 is 10200. This number meets all the requirements and can be used for any calculations related to divisibility by these numbers!