Introduction

The objective of this article is to explore the process of determining the least six-digit number that is exactly divisible by 15, 20, and 25. This topic is crucial for understanding the concept of the least common multiple (LCM) and its application in solving arithmetic problems. We will also cover the steps of prime factorization and how to calculate LCM, as well as practical examples to solidify the concepts.

Prime Factorization and LCM Calculation

To solve this problem, we first need to perform the prime factorization of each of the numbers involved—15, 20, and 25. Prime factorization is the process of breaking down a number into its prime factors, which are the smallest natural numbers that, when multiplied together, give the original number.

Step 1: Prime Factorization

Let's start with the prime factorization of each number:

15: 15 3 × 5 20: 20 2^2 × 5 25: 25 5^2

Step 2: Determine the LCM

The least common multiple (LCM) is the smallest number that is a multiple of all the given numbers. To determine the LCM, we take the highest power of each prime number that appears in the factorizations. This gives us:

For the prime factor 2: The highest power of 2 is 2^2, from 20. For the prime factor 3: The highest power of 3 is 3^1, from 15. For the prime factor 5: The highest power of 5 is 5^2, from 25.

Therefore, the LCM is:

LCM 2^2 × 3^1 × 5^2 4 × 3 × 25 300

Finding the Least Six-Digit Number

The smallest six-digit number is 100,000. To find the smallest six-digit number that is divisible by 300, we need to follow these steps:

Divide 100,000 by 300: ( frac{100,000}{300} approx 333.33 ) Rounding up: The rounded-up value is 334. This is because we need to find the smallest whole number that is greater than or equal to 333.33. Multiply by 300: 334 × 300 100,200

The least six-digit number that is exactly divisible by 15, 20, and 25 is 100,200.

Additional Examples

Let's consider another example to further understand the concept:

First, find the LCM of 15, 20, and 25. Perform prime factorization: 15 3 × 5 20 2^2 × 5 25 5^2 Find the highest power of each prime factor: 2^2 3^1 5^2 Calculate the LCM: 2^2 × 3^1 × 5^2 300. To find the smallest 6-digit number, take the ceiling of ( frac{100,000}{300} ) and multiply by 300. ( lceil frac{100,000}{300} rceil 334 ), so the least six-digit number is 334 × 300 102,000.

These examples and explanations should help clarify the steps involved in finding the LCM and determining the least six-digit number divisible by the given numbers.