Problem 26 - Fundamentals of Mathematics

Question

Find the least common multiple of the numbers. 3 and 4

Step-by-Step Solution

Answer

The least common multiple of 3 and 4 is 12.

Step 1: Understand the Definition of LCM

The least common multiple (LCM) of two numbers is the smallest positive integer that is divisible by both numbers. To find the LCM, you look for the smallest common multiple shared by the two numbers.

Step 2: List the Multiples of Each Number

Begin by listing some multiples of each number. - Multiples of 3: 3, 6, 9, 12, 15, 18, ... - Multiples of 4: 4, 8, 12, 16, 20, ...

Step 3: Identify the Common Multiples

From the lists created in Step 2, identify the multiples that are common to both numbers. In this case, observe the multiples: - Common multiples: 12, ...

Step 4: Determine the Least Common Multiple

Once you've identified the common multiples, find the smallest value. From the identified common multiples, the smallest one is 12. Therefore, the LCM is 12.

Key Concepts

Multiples

In number theory, a **multiple** of a number is simply the product of that number and any integer. For example, the multiples of 3 are found by multiplying 3 by integers: 3, 6, 9, 12, and so on. Similarly, the multiples of 4 are 4, 8, 12, 16, etc.

• Multiplying by 1 gives the original number, so every number is a multiple of itself.
• Multiples extend infinitely, as you can continue multiplying by ever larger integers.
• Understanding multiples is crucial in many areas of mathematics, including finding common multiples and determining least common multiples.

This foundational knowledge helps in operations like simplifying fractions and finding patterns in math.

Common Multiple

A **common multiple** is a number that two or more numbers share as a multiple. To find a common multiple, we list the multiples of the numbers involved and identify the ones both lists have.
For instance, to find common multiples of 3 and 4:

• The multiples of 3 are: 3, 6, 9, 12, 15...
• The multiples of 4 are: 4, 8, 12, 16, 20...

The common multiple they share is 12 in this example.
The concept of common multiples is extremely useful when performing operations on fractions, like finding least common denominators for addition or subtraction of fractions. It also plays a role in more advanced topics like solving simultaneous equations.

Number Theory

**Number theory** is a branch of mathematics that explores the properties and relationships of numbers, especially integers. This area of mathematics involves understanding concepts like divisibility, prime numbers, and integer operations.
Number theory helps us understand:

• The structure of integers, including how they can be expressed as products of primes.
• Why certain numbers behave the way they do, according to their divisors and multiples.

When you find the least common multiple, you are engaging with number theory to some extent. It requires a comprehension of how numbers interact, share properties like common multiples, and what makes certain numbers unique.