What is the least number divisible by 30, 45, and 15 exactly?

The least number exactly divisible by 30, 45, and 15 is 90.

Here's how to find the Least Common Multiple (LCM), which represents the smallest number divisible by a set of numbers:

Understanding the Problem

We need to find the smallest number that all three numbers (30, 45, and 15) can divide into without leaving a remainder. This is the Least Common Multiple (LCM).

Methods to Find the LCM

Several methods can be used to calculate the LCM. Here are two common approaches:

• Listing Multiples: List the multiples of each number until a common multiple is found. This can be time-consuming for larger numbers.
• Prime Factorization: This is the most efficient and reliable method.

Using Prime Factorization

1. Find the Prime Factorization of Each Number:

   • 30 = 2 x 3 x 5
   • 45 = 3 x 3 x 5 = 3² x 5
   • 15 = 3 x 5

2. Identify the Highest Power of Each Prime Factor:

   • 2¹ (from 30)
   • 3² (from 45)
   • 5¹ (from 30, 45, and 15)

3. Multiply the Highest Powers Together:

   • LCM = 2¹ x 3² x 5¹ = 2 x 9 x 5 = 90

Therefore, the least common multiple of 30, 45, and 15 is 90. This means 90 is the smallest number that can be divided evenly by all three numbers.