What are Common Factors?

S3-SA4-0286

Grade Level:

Class 6

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

Definition

What is it?

Common factors are numbers that divide two or more given numbers exactly, without leaving any remainder. Think of them as shared 'dividers' for a group of numbers. For example, if you have two numbers, a common factor is a number that is a factor of both of them.

Simple Example

Quick Example

Imagine you have 12 ladoos and your friend has 18 ladoos. You want to divide both sets of ladoos equally into smaller plates. The numbers of ladoos that can be put on each plate (like 1, 2, 3, 6) are common factors because they divide both 12 and 18 exactly.

Worked Example

Step-by-Step

Let's find the common factors of 10 and 15.

Step 1: Find all factors of 10.
Factors of 10 are the numbers that divide 10 exactly: 1, 2, 5, 10.

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Step 2: Find all factors of 15.
Factors of 15 are the numbers that divide 15 exactly: 1, 3, 5, 15.

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Step 3: Compare the lists of factors.
Factors of 10: {1, 2, 5, 10}
Factors of 15: {1, 3, 5, 15}

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Step 4: Identify the numbers that appear in both lists.
The numbers that are common to both lists are 1 and 5.

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Answer: The common factors of 10 and 15 are 1 and 5.

Why It Matters

Understanding common factors helps engineers design efficient systems and computer scientists optimize code. It's a foundational skill for solving problems in areas like data encryption in cryptography and even predicting patterns in economics. This simple concept is used by experts to build secure apps and powerful AI models!

Common Mistakes

MISTAKE: Listing only some factors, not all, for a number. | CORRECTION: Always find all factors of each number first, by systematically checking numbers from 1 up to the number itself.

MISTAKE: Confusing factors with multiples. | CORRECTION: Factors are numbers that divide a given number exactly (smaller or equal). Multiples are numbers you get when you multiply a given number by other whole numbers (larger or equal).

MISTAKE: Forgetting that 1 is always a common factor. | CORRECTION: Remember that 1 divides every number exactly, so 1 will always be a common factor for any set of numbers.

Practice Questions

Try It Yourself

QUESTION: Find the common factors of 6 and 9. | ANSWER: 1, 3

QUESTION: What are the common factors of 12, 18, and 24? | ANSWER: 1, 2, 3, 6

QUESTION: A shopkeeper has 20 pens and 30 pencils. She wants to pack them into identical gift boxes, with no items left over. What are the possible number of pens/pencils she can put in each box? (Hint: Find the common factors). | ANSWER: 1, 2, 5, 10

MCQ

Quick Quiz

Which of the following is NOT a common factor of 16 and 20?

The Correct Answer Is:

The factors of 16 are 1, 2, 4, 8, 16. The factors of 20 are 1, 2, 4, 5, 10, 20. The common factors are 1, 2, 4. So, 8 is not a common factor of 16 and 20.

Real World Connection

In the Real World

Imagine a chef in a busy restaurant in Delhi needs to divide 36 rotis and 48 parathas equally among several serving trays. To find the largest number of rotis/parathas that can go on each tray without any leftover, the chef would use common factors (specifically, the highest common factor). This helps in efficient meal prep and serving.

Key Vocabulary

Key Terms

FACTOR: A number that divides another number exactly, leaving no remainder. | DIVIDE: To split a number into equal parts. | REMAINDER: The amount left over after division. | COMMON: Shared by two or more things.

What's Next

What to Learn Next

Great job understanding common factors! The next step is to learn about the 'Highest Common Factor' (HCF). It's the biggest common factor, and it's super useful for solving many real-life problems, like dividing things into the largest possible equal groups.