What is Common Multiples?

S3-SA4-0029

Grade Level:

Class 6

AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering

Definition

What is it?

Common Multiples are numbers that are multiples of two or more different numbers at the same time. Think of them as numbers that appear in the multiplication tables of multiple numbers simultaneously.

Simple Example

Quick Example

Imagine you have two friends, Rohan and Priya. Rohan buys a new cricket bat every 2 months, and Priya buys one every 3 months. In which months will they both buy a new bat? The months would be multiples of both 2 and 3, like 6 months, 12 months, and so on. These are their common multiples.

Worked Example

Step-by-Step

Let's find the first three common multiples of 4 and 6.

1. List the multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, 36, ...
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2. List the multiples of 6: 6, 12, 18, 24, 30, 36, 42, ...
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3. Now, look for numbers that appear in BOTH lists.
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4. The first number appearing in both lists is 12.
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5. The next number appearing in both lists is 24.
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6. The third number appearing in both lists is 36.
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Answer: The first three common multiples of 4 and 6 are 12, 24, and 36.

Why It Matters

Understanding common multiples helps us in many areas, from scheduling tasks efficiently in project management to designing efficient computer algorithms. Engineers use this concept to synchronize different parts of a machine, and even in data science, it helps in pattern recognition. It's a foundational skill for future innovators!

Common Mistakes

MISTAKE: Confusing common multiples with common factors. | CORRECTION: Multiples are products (numbers you get by multiplying), while factors are numbers that divide evenly into another number.

MISTAKE: Stopping after finding only the first common multiple. | CORRECTION: Remember that common multiples are infinite. Always read the question carefully to see how many common multiples are asked for.

MISTAKE: Not listing enough multiples to find the common ones. | CORRECTION: Always list a good number of multiples for each number first, then compare the lists to find the common ones.

Practice Questions

Try It Yourself

QUESTION: Find the first common multiple of 5 and 10. | ANSWER: 10

QUESTION: List the first two common multiples of 3 and 4. | ANSWER: 12, 24

QUESTION: A bus from City A leaves every 8 hours, and a bus from City B leaves every 12 hours. If they both left at 6 AM today, after how many hours will they both leave at the same time again? | ANSWER: 24 hours

MCQ

Quick Quiz

Which of the following is a common multiple of 2, 3, and 4?

The Correct Answer Is:

12 is a multiple of 2 (2x6), a multiple of 3 (3x4), and a multiple of 4 (4x3). The other options are not multiples of all three numbers.

Real World Connection

In the Real World

Imagine a traffic light system in a busy Indian city. Different lights might change every 30 seconds, 45 seconds, and 60 seconds. To program them to all turn green or red at the exact same moment for smooth traffic flow, engineers need to find their common multiples. This ensures synchronized movement and reduces traffic jams.

Key Vocabulary

Key Terms

MULTIPLE: The result of multiplying a number by another whole number. | FACTOR: A number that divides another number exactly, without leaving a remainder. | COMMON: Shared by two or more things. | INFINITE: Having no end or limit.

What's Next

What to Learn Next

Great job understanding Common Multiples! Now that you know what they are, the next exciting step is to learn about the 'Least Common Multiple' (LCM). It builds directly on this concept and is super useful for adding and subtracting fractions!