Problem 45 - Fundamentals of Mathematics

Question

Find the least common multiple of the numbers. \(4,5,\) and 21

Step-by-Step Solution

Answer

The least common multiple is 420.

Step 1: List Prime Factors

Start by finding the prime factorization of each number. For 4, the prime factorization is \(2^2\). For 5, it is \(5^1\). For 21, it is \(3^1 \times 7^1\).

Step 2: Identify the Maximum Exponent for Each Prime

List all prime numbers found in any of the factorizations and determine the highest power for each. The primes we have are 2, 3, 5, and 7. The highest powers are: 2 appears as \(2^2\), 3 appears as \(3^1\), 5 appears as \(5^1\), and 7 appears as \(7^1\).

Step 3: Calculate the Least Common Multiple

The least common multiple (LCM) is the product of each unique prime raised to its highest power found in the factorizations. Multiply: \(2^2, 3^1, 5^1, 7^1\). Thus, the LCM is \(4 \times 3 \times 5 \times 7\).

Step 4: Perform the Multiplication

First, multiply 4 by 3: \(4 \times 3 = 12\). Then multiply the result by 5: \(12 \times 5 = 60\). Finally, multiply by 7: \(60 \times 7 = 420\).

Key Concepts

Prime Factorization

Prime factorization is the process of breaking down a composite number into its basic building blocks, which are prime numbers. A composite number is any number that can be divided exactly by numbers other than 1 and itself. Prime numbers, on the other hand, are numbers greater than 1 that have no divisors other than 1 and themselves.
For example, to perform prime factorization on the number 4, we divide it into 2 and 2, resulting in the expression \(2^2\). Similarly, the number 5 is already a prime number, so its factorization is simply \(5^1\). For the number 21, we find that it is divisible by 3, resulting in 7, both of which are primes, so the factorization is \(3^1 \times 7^1\).

• Prime factorization simplifies understanding the structure of numbers.
• It helps to simplify mathematics operations involving multiple numbers, such as finding the least common multiple.

Breaking numbers into their prime factors is a fundamental step in solving many numerical problems.

Prime Numbers

Prime numbers are interesting entities in mathematics because they have exactly two distinct positive divisors: 1 and the number itself. The list of prime numbers begins with 2, 3, 5, 7, and continues indefinitely. They are considered the building blocks of the number system, similar to atoms in chemistry. Each non-prime can be built as a product of primes.
Understanding prime numbers is crucial, especially in the context of problems like finding the least common multiple. In our exercise, numbers like 4, 5, and 21 were broken down to their core prime factors. A useful tip is to always check divisibility starting from the smallest prime: 2, then 3, 5, and so on.
Knowing these numbers and how to identify them helps increase mathematical intuition. This is especially important with complex divisions and when exploring deeper relationships within numbers.

Multiplication

Multiplication is one of the most fundamental operations in mathematics, which involves calculating the total of one number taken a certain number of times. It can be visualized easily with integers, such as calculating the area of a rectangle, or abstractly, with prime factor exponents as in our exercise.
To find the least common multiple (LCM), multiplication involves combining the prime factors of the numbers involved, each raised to their highest power among the numbers. In simpler terms, the LCM is obtained by multiplying together all the prime factors discovered previously, with their highest exponents, as seen here:

• We first multiply the largest power of each prime: \(2^2\), \(3^1\), \(5^1\), \(7^1\).
• Calculating progressively: \( 4 \times 3 = 12 \), then \( 12 \times 5 = 60 \), and finally \( 60 \times 7 = 420 \).

Mastering multiplication not only helps in finding the LCM but is fundamental to various aspects of daily arithmetic and problem-solving.