Problem 60 - Fundamentals of Mathematics

Question

Find the least common multiple of the numbers. \(3,9,12,\) and 3

Step-by-Step Solution

Answer

The least common multiple is 36.

Step 1: Identify the greatest number

First, look at all the numbers: 3, 9, 12, and 3. The greatest number here is 12. Finding the least common multiple (LCM) often involves the greatest number, as it may be one of the multiples.

Step 2: List multiples of the greatest number

List some multiples of 12, since it's the largest number. These are 12, 24, 36, 48, 60, and so on. Note these multiples, as the LCM will be one of these.

Step 3: Check divisibility by the other numbers

Now take each multiple of 12 and check if it is divisible by 3, 9, and 3. If a multiple is divisible by all of these numbers, it is a common multiple.

Step 4: Determine the smallest common multiple

Start with the smallest multiple of 12, which is 12 itself. Check if it is divisible by 3 and 9. 12/3 = 4, whole number. 12/9 = 1 with a remainder. So 12 is not divisible by all terms. Next, try 24. 24/3 = 8, whole number. 24/9 = 2 with a remainder, so move on. Next is 36. 36/3 = 12, whole number. 36/9 = 4, whole number. Since 36 is divisible by 3 and 9, it is our candidate for the least common multiple.

Step 5: Confirm the LCM

Check that 36 is also divisible by both occurrences of 3 (which it is, since 36 is divisible by any number that divides 3 as well). Therefore, the smallest number that is divisible by all terms (3, 9, 12) is 36, confirming it as the LCM.

Key Concepts

Divisibility

Divisibility is the ability of one number to be evenly divided by another. A number is said to be divisible by another if the division results in a whole number without any remainder. Understanding divisibility is essential when finding the least common multiple (LCM) because it helps determine if one number can fit perfectly into another without a leftover.

• For example, 12 is divisible by 3 because when you divide 12 by 3, the result is 4, a whole number.
• However, if you divide 15 by 4, the result is not a whole number, indicating 15 is not divisible by 4.

Determining divisibility involves simple division of the numbers and checking if the quotient is an integer. This concept plays a crucial part in identifying common multiples. It allows us to see which numbers share the same factors.

Multiples

Multiples of a number are the products obtained when that number is multiplied by integers. Understanding multiples is key to finding the least common multiple because it allows us to identify numbers that are common to different lists of multiples.

• For example, the multiples of 12 include 12, 24, 36, 48, and so forth.
• Similarly, the multiples of 3 are 3, 6, 9, 12, 15, and so on.

To find the LCM of a set of numbers, list the multiples of each number until you find the smallest multiple that is common to each list. Multiples help identify these shared numbers efficiently.

Greatest Number

The greatest number in a given set of numbers is significant when determining the least common multiple, as it serves as a starting point for listed multiples. The greatest number, being the largest, often means that its multiples are larger as well.

• In the exercise, 12 is the greatest of the numbers 3, 9, 12, and 3.
• Listing the multiples of the greatest number provides an organized way to approach finding the LCM.

By focusing on the greatest number's multiples first, you can reduce the number of calculations needed to find a common multiple for the other numbers, saving time and simplifying the process.