Problem 27 - Fundamentals of Mathematics

Question

Find the least common multiple of the numbers. 2 and 9

Step-by-Step Solution

Answer

The LCM of 2 and 9 is 18.

Step 1: Understand the Problem

In this exercise, we need to find the least common multiple (LCM) of the numbers 2 and 9. The LCM of two numbers is the smallest positive integer that is divisible by both numbers.

Step 2: List the Multiples

Start by listing some multiples of each number. - Multiples of 2: 2, 4, 6, 8, 10, 12, 14, 16, 18... - Multiples of 9: 9, 18, 27, 36... The aim is to find the smallest multiple they have in common.

Step 3: Identify the Common Multiples

Compare the lists of multiples from Step 2 to determine if they have any shared values. - For the lists provided, the first number they share is 18.

Step 4: Conclusion

Let's conclude that the least common multiple of 2 and 9 is the smallest number found in both lists of multiples. From the previous steps, we determined this shared value to be 18.

Key Concepts

Multiples

When you're beginning to explore the world of numbers, understanding the concept of "multiples" is key. A multiple of a number is what you obtain when you multiply that number by an integer. For example, when we talk about multiples of 2, we mean: 2, 4, 6, 8, 10, and so on.
These are all produced by multiplying 2 with successive whole numbers like 1, 2, 3, etc. You might think of it like jumping forward the same number each time on a number line.

• 2 multiplied by 1 is 2
• 2 multiplied by 2 is 4
• 2 multiplied by 3 is 6

Similarly, multiples of 9 would be 9, 18, 27, and so forth.
This repetitive addition is fundamental in understanding how numbers relate to one another.

Common Multiples

"Common multiples" are simply those multiples that two or more numbers share. When two lists of multiples overlap, those overlapping numbers are called common multiples.
Let's delve into the lists of multiples for our example numbers:

• Multiples of 2: 2, 4, 6, 8, 10, 12, 14, 16, 18...
• Multiples of 9: 9, 18, 27, 36...

When you look for common ground in these lists, you find they both include the number 18.
This is why 18 pops up as a common multiple of both 2 and 9. Finding these common numbers is the foundation for calculating the least common multiple (LCM), the smallest positive number in both lists.

Divisibility

"Divisibility" is at the heart of understanding least common multiples. A number is divisible by another if it divides without leaving a remainder. To find out if 18 is truly the least common multiple of 2 and 9, check its divisibility by both numbers.

• 18 divided by 2 equals 9, and it goes evenly, so 18 is divisible by 2.
• 18 divided by 9 equals 2, with no remainder, so 18 is also divisible by 9.

Because 18 divides perfectly by both 2 and 9, it is confirmed as the least common multiple.
Divisibility is an essential concept because it ensures that the number we choose as the least common multiple genuinely works with all given numbers, making it a fail-safe choice in math.