Problem 22 - Fundamentals of Mathematics

Question

Find the least common multiple of the numbers. 9 and 18

Step-by-Step Solution

Answer

The least common multiple of 9 and 18 is 18.

Step 1: Define Least Common Multiple (LCM)

The Least Common Multiple (LCM) of two numbers is the smallest number that is a multiple of both numbers. In this exercise, we need to find the smallest number that both 9 and 18 can divide into without a remainder.

Step 2: List Multiples

List the multiples of each number to identify the common ones. For 9, the multiples are 9, 18, 27, 36, etc. For 18, the multiples are 18, 36, 54, etc.

Step 3: Identify the Common Multiples

Compare the lists of multiples from Step 2. The common multiples of 9 and 18 from the lists are 18, 36, etc.

Step 4: Choose the Smallest Common Multiple

From the common multiples identified in Step 3, select the smallest one. Both lists have 18 as the smallest common multiple.

Key Concepts

Understanding Multiples

In mathematics, a multiple of a number is simply the product of that number and any integer. This means if you take a number, say 9, and multiply it by 1, 2, 3, and so on, you will get multiples of 9. For example:

• 9 × 1 = 9
• 9 × 2 = 18
• 9 × 3 = 27

This process applies to any number, allowing us to list out its multiples easily. Multiples provide us with a basis to explore further arithmetic concepts, such as dividing, factoring, and finding commonality between numbers in terms of divisibility issues.

Concept of Common Multiples

Common multiples are the numbers that appear in the list of multiples of two or more numbers. These can be identified by comparing the multiple lists of the numbers involved. For example, when comparing the multiples of 9 and 18:

• Multiples of 9: 9, 18, 27, 36, ...
• Multiples of 18: 18, 36, 54, ...

You can see that 18 and 36 are multiples common to both. Finding common multiples is essential when determining the least common multiple since it helps establish a number that two values can fit into without any remainders. This forms the basis for solving problems involving LCM, or the least common multiple.

Divisibility Rules

Divisibility is an important concept when finding the least common multiple (LCM). A number is considered divisible by another if you divide them, and the result is a whole number without any remainder. For instance, 18 is divisible by 9 because 18 divided by 9 equals 2, which is an integer. This lack of remainder when dividing is a sign of divisibility.
Divisibility rules are simple guidelines used to quickly determine whether a number can be divided by another without actually doing the division. For example, any even number is divisible by 2, and any number ending in 0 or 5 is divisible by 5. Understanding these rules can make it easier to find the least common multiple of a set of numbers, as it helps identify numbers that both original numbers can divide into evenly.