What are Common Multiples?

S3-SA4-0287

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.

Simple Example

Quick Example

Imagine you have two friends, Rohan and Priya. Rohan likes to eat a snack every 2 hours, and Priya likes to eat a snack every 3 hours. When will they both be eating a snack at the same time? They will both eat at 6 hours, 12 hours, 18 hours, and so on. These are the common multiples of 2 and 3.

Worked Example

Step-by-Step

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

Step 1: List the first few multiples of 4.
Multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, 36, ...

---Step 2: List the first few multiples of 6.
Multiples of 6: 6, 12, 18, 24, 30, 36, 42, ...

---Step 3: Look for numbers that appear in both lists.
Common numbers are 12, 24, 36.

---Step 4: Identify the first three common multiples.
The first three common multiples of 4 and 6 are 12, 24, and 36.

Why It Matters

Understanding common multiples helps in fields like Computer Science for scheduling tasks or in Physics for understanding cycles. Engineers use this concept when designing gears or setting up production lines efficiently. It's a foundational skill for solving many real-world problems.

Common Mistakes

MISTAKE: Confusing common multiples with common factors. | CORRECTION: Multiples are products (multiplication), factors are numbers that divide evenly (division). Multiples are generally larger than or equal to the original numbers, while factors are smaller or equal.

MISTAKE: Stopping after finding only the first common multiple. | CORRECTION: There are infinitely many common multiples for any two numbers. The question usually asks for a specific number of common multiples (e.g., 'first three') or the 'least common multiple'.

MISTAKE: Listing multiples incorrectly by missing numbers. | CORRECTION: Always list multiples systematically by continuously adding the original number (e.g., for 5: 5, 5+5=10, 10+5=15, etc.) or by multiplying it by 1, 2, 3, and so on.

Practice Questions

Try It Yourself

QUESTION: Find the first two common multiples of 3 and 5. | ANSWER: 15, 30

QUESTION: Which of these numbers is a common multiple of 2, 4, and 8: 12, 16, 20? | ANSWER: 16

QUESTION: A bus leaves from Stand A every 6 minutes, and another bus leaves from Stand B every 9 minutes. If both left at 9:00 AM, when will they both leave at the same time again? | ANSWER: 9:18 AM

MCQ

Quick Quiz

Which of the following is a common multiple of 5 and 7?

The Correct Answer Is:

35 is a multiple of 5 (5 x 7 = 35) and also a multiple of 7 (7 x 5 = 35). The other options are only multiples of one number or neither.

Real World Connection

In the Real World

Imagine a traffic signal system in a smart city. To ensure smooth flow, traffic lights are programmed to change colors at specific intervals. Engineers use common multiples to synchronize these lights so that green signals align at important intersections, preventing jams and making your daily commute smoother.

Key Vocabulary

Key Terms

MULTIPLE: The result of multiplying a number by an integer. E.g., 10 is a multiple of 5. | COMMON: Shared by two or more things. | LEAST COMMON MULTIPLE (LCM): The smallest positive common multiple of two or more numbers. | FACTOR: A number that divides another number exactly, without leaving a remainder. | PRODUCT: The result of multiplication.

What's Next

What to Learn Next

Great job understanding common multiples! Next, you should explore the 'Least Common Multiple (LCM)'. It's the smallest of all common multiples and is super useful for solving problems involving fractions and scheduling.