91Ó°ÊÓ

Prime factorization involves breaking down a number into its prime number factors. A prime number is a number greater than 1 that has no positive divisors other than 1 and itself.
To factorize a number, start by dividing it by the smallest prime number (2, 3, 5, etc.). Continue dividing the quotient by prime numbers until you get a quotient of 1.
For example, to prime factorize 10 and 25:
10 can be divided by 2 (the smallest prime number), giving a quotient of 5. Since 5 is already a prime, you stop there. Hence, 10 = 2 × 5.
For 25, since it is not divisible by 2 or 3, we move to the next prime number, which is 5. 25 divided by 5 gives 5, and another division by 5 results in 1. Therefore, 25 = 5 × 5 = 5^2.
This step is crucial because it helps in identifying the prime factors which are used to calculate the LCM later.

The Least Common Multiple (LCM) of two or more numbers is the smallest number that is a multiple of all the numbers. It can be found using the prime factorization of the numbers involved.
The steps to calculate the LCM are straightforward:

• Prime factorize all numbers.
• Identify the highest power of each prime factor present in the factorization.
• Multiply these highest powers to get the LCM.

For instance, for the numbers 10 and 25:
10 = 2^1 × 5^1
25 = 5^2
The prime factors are 2 and 5. The highest power of 2 is 2^1, and the highest power of 5 is 5^2. Multiplying these highest powers gives us the LCM:
LCM = 2^1 × 5^2 = 2 × 25 = 50.
So, the least common multiple of 10 and 25 is 50.

Understanding how to follow mathematics steps is essential for solving problems like finding the LCM. Each step has its purpose, and here's a simplified breakdown:
Step 1: Perform Prime Factorization.

• This step involves breaking down each number into its basic prime factors. It sets the stage for identifying common factors.

For example, 10 = 2 × 5 and 25 = 5 × 5 = 5^2.
Step 2: Identify the Highest Powers.

• From the prime factorizations, find the highest power of each prime number.
• This ensures that the LCM is a multiple of each number involved.

For the numbers 10 and 25, the highest power of 2 is 2^1 and of 5 is 5^2.
Step 3: Multiply the Highest Powers.

• Finally, multiply these highest powers to get the LCM.

By multiplying 2^1 and 5^2, we get 50, which is the LCM.
These steps are simple and structured, making it easier to find the LCM for any set of numbers.