Problem 38 - Algebra A Combined Function

Question

Objective C Find the LCM of each list of numbers. See Examples 7 through 9 . 9,15

Step-by-Step Solution

Answer

The LCM of 9 and 15 is 45.

Step 1: Prime Factorize Each Number

Start by finding the prime factors of each number in the list. For 9, the prime factorization is \( 9 = 3 \times 3 = 3^2 \). For 15, the prime factorization is \( 15 = 3 \times 5 \).

Step 2: Identify Common and Unique Factors

List all unique factors from both numbers. We have 3 and 5 as the base factors. Take each factor to the highest power it appears in either factorization.

Step 3: Determine the LCM

The least common multiple (LCM) is found by multiplying the highest power of each factor identified. For 3, the highest power is \( 3^2 \) and for 5, it is \( 5^1 \). Thus, the LCM is \( 3^2 \times 5 = 9 \times 5 = 45 \).

Key Concepts

Prime Factorization

Prime factorization is a key concept in finding the Least Common Multiple (LCM) of numbers. It involves breaking down a number into its basic building blocks, which are prime numbers. A prime number is any number greater than 1 that cannot be divided evenly by any other numbers except for 1 and itself. For example, in the original exercise, the numbers 9 and 15 were broken down into their prime factors.

• The number 9 was factorized into two 3s, or mathematically expressed as: \(9 = 3 \times 3 = 3^2\).
• The number 15 was factorized into 3 and 5, or: \(15 = 3 \times 5\).

Doing this helps us identify the shared factors and those unique to each number. By understanding prime factorization, it becomes easier to solve a variety of number theory problems, including finding the LCM.

Unique Factors

Identifying unique factors is an important step in calculating the LCM. After prime factorization, we list all different prime numbers found in the given numbers. Unique factors are the primes that appear in any factorization.
In this example:

• The unique prime factors of 9 and 15 are 3 and 5.

To find the LCM, each of these unique factors needs to be considered and taken to the highest power it appears in any of the original numbers’ factorizations.
It’s this collection of unique factors, taken with their highest powers, that allows the calculation of the least common multiple.

Number Theory

Number theory is a fascinating branch of mathematics focusing on the properties of numbers, especially integers. It explores concepts such as divisibility, prime numbers, and integer solutions. Understanding number theory helps us solve problems involving factorization, greatest common divisors, and least common multiples.
The LCM, an essential concept in number theory, helps us deal with problems involving repeated patterns or coinciding events. By determining the LCM, we find the smallest number divisible by each number in a set, relevant in various applications like scheduling or syncing cycles.
Practicing these concepts in exercises enhances understanding and builds foundational mathematical skills.

Mathematical Operations

In finding the LCM, mathematical operations such as multiplication play a significant role. Once unique factors and their powers are determined, multiplying them gives the LCM. This step uses the mathematical operation of exponentiation before simplification through multiplication.
For the numbers provided in the exercise:

• The highest power of 3 is \(3^2\).
• The highest power of 5 is \(5^1\).
• Multiplying these, we have the LCM: \(3^2 \times 5 = 9 \times 5 = 45\).

Such operations illustrate the use of basic arithmetic in problem solving. Mastery of these operations is central to working with more complex mathematical concepts and real-world applications.