Problem 28 - Fundamentals of Mathematics

Question

Find the least common multiple of the numbers. 7 and 9

Step-by-Step Solution

Answer

The LCM of 7 and 9 is 63.

Step 1: Understanding LCM

The least common multiple (LCM) of two numbers is the smallest number that is a multiple of both numbers. For two numbers, you look for the lowest number that both numbers divide into without leaving a remainder.

Step 2: List the Multiples of Each Number

List the multiples of 7: 7, 14, 21, 28, 35, 42, 49, 56, 63, 70, 77, 84, ... List the multiples of 9: 9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 99, 108, ...

Step 3: Find the Common Multiples

Look for common multiples in the two lists. From the listed multiples of 7 and 9, we can see that 63 is a common multiple.

Step 4: Identify the Least Common Multiple

Among the common multiples found, identify the smallest one. In this case, the smallest common multiple is 63.

Key Concepts

Multiples of 7

When we talk about multiples of 7, we're essentially looking at numbers that result from multiplying 7 by any whole number. Think of them as stepping stones created by repeatedly adding 7 to itself. Starting from the beginning:

• The first multiple of 7 is 7 itself, as 7 multiplied by 1 is 7.
• The second multiple is 14, because 7 multiplied by 2 equals 14.
• Continuing this pattern, you get 21, 28, 35, and so on.

By understanding these multiples, you can easily generate a sequence: 7, 14, 21, 28, 35, 42, 49, 56, 63, 70, 77, and beyond. Just pick any positive integer, multiply it by 7, and voilà, you have another multiple!
This set of multiples proves handy when solving for the least common multiple of any two numbers, as we'll soon see.

Multiples of 9

Just like with the multiples of 7, multiples of 9 are all about stretching the number 9 by multiplying it with whole numbers. To break it down simply, multiples of 9 look like this:

• The first multiple is 9 (as 9 times 1 equals 9).
• The next one is 18 (9 times 2).
• Then 27 follows (9 times 3), and so on.

Other numbers in this chain include 36, 45, 54, 63, 72, and more, just by continuing this multiplication process. This sequence forms a nice pattern that we can use to identify common multiples, an essential step in finding the least common multiple for numbers like 7 and 9.
Understanding these sequences makes the task of finding common multiples much easier, as we'll explain next.

Identifying Common Multiples

Finding common multiples can feel like a search for hidden treasures in two sequences of numbers. The trick is to line up the multiples from each number and spot the ones they share. Let's dive in with our specific example of 7 and 9.

• For 7, you’ve got multiples like 7, 14, 21, 28, 35, 42, 49, 56, 63, etc.
• For 9, you’ll list 9, 18, 27, 36, 45, 54, 63, etc.

To identify common multiples, you need to find numbers appearing in both lists. In this instance, 63 catches our eye. It's the first number showing up in both sequences, making it a common multiple.
Once we've identified 63 as common, we specifically note it's the smallest common one, helping us define it as the least common multiple (LCM). This means, 63 is the smallest number that both 7 and 9 will divide into completely, without any remainder. And there you have it! Just like that, we've unraveled the mystery of identifying common multiples, sharpening our skills for all future LCM problems.