What is Divisibility by Prime Factors? | Simple Explanation for Class 8

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What is the Concept of Divisibility by Prime Factors?

Grade Level:

Class 8

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

Definition

What is it?

The concept of divisibility by prime factors helps us understand if a number can be perfectly divided by another number by just looking at their prime building blocks. It states that if a number 'A' is divisible by another number 'B', then all the prime factors of 'B' must also be present in the prime factors of 'A'.

Simple Example

Quick Example

Imagine you have 12 ladoos and you want to share them equally among 4 friends. You can do it because 12 is divisible by 4. If we break them down, the prime factors of 4 are 2 and 2. The prime factors of 12 are 2, 2, and 3. See how the prime factors of 4 (two 2s) are all present in the prime factors of 12? This is exactly what the concept means!

Worked Example

Step-by-Step

Let's check if 60 is divisible by 12 using prime factors.
1. Find the prime factors of 60.
60 = 2 x 30
60 = 2 x 2 x 15
60 = 2 x 2 x 3 x 5
So, the prime factors of 60 are {2, 2, 3, 5}.
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2. Find the prime factors of 12.
12 = 2 x 6
12 = 2 x 2 x 3
So, the prime factors of 12 are {2, 2, 3}.
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3. Compare the prime factors.
Are all the prime factors of 12 (two 2s and one 3) present in the prime factors of 60 (two 2s, one 3, and one 5)? Yes, they are!
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4. Conclusion: Since all prime factors of 12 are present in 60, 60 is divisible by 12. (60 / 12 = 5)

Why It Matters

Understanding divisibility by prime factors is super important in many advanced fields. In Computer Science, it's used in algorithms for data encryption and security, like protecting your WhatsApp messages. Engineers use it for designing efficient systems, and Data Scientists use it to analyze patterns in large datasets, helping companies understand customer behavior.

Common Mistakes

MISTAKE: Thinking that if a prime factor of the divisor is present, it's enough (e.g., 10 is divisible by 4 because both have a '2'). | CORRECTION: ALL prime factors of the divisor must be present in the dividend, with at least the same frequency.

MISTAKE: Confusing prime factors with just 'factors'. (e.g., saying 6 is a prime factor of 12). | CORRECTION: Prime factors must be prime numbers themselves. 6 is a factor of 12, but its prime factors are 2 and 3.

MISTAKE: Not finding all prime factors correctly, especially for larger numbers. (e.g., saying 20 = 2 x 10 is fully prime factorized). | CORRECTION: Always break down composite factors until only prime numbers remain. 20 = 2 x 2 x 5.

Practice Questions

Try It Yourself

QUESTION: Is 45 divisible by 9 using the concept of prime factors? | ANSWER: Yes

QUESTION: Using prime factors, determine if 84 is divisible by 14. | ANSWER: Yes

QUESTION: A shopkeeper has 108 pencils. Can he make packs of 18 pencils each without any left over? Explain using prime factors. | ANSWER: Yes. Prime factors of 18 are {2, 3, 3}. Prime factors of 108 are {2, 2, 3, 3, 3}. All prime factors of 18 are present in 108.

MCQ

Quick Quiz

Which of the following statements is true about the divisibility of 72 by 24?

The prime factors of 72 are {2, 2, 2, 3} and the prime factors of 24 are {2, 2, 3}.

The prime factors of 24 are {2, 2, 2, 3} and the prime factors of 72 are {2, 2, 3, 3}.

The prime factors of 72 are {2, 2, 2, 3, 3} and the prime factors of 24 are {2, 2, 2, 3}. All factors of 24 are in 72.

The prime factors of 72 are {2, 3, 3} and the prime factors of 24 are {2, 2, 2}.

The Correct Answer Is:

Option C correctly lists the prime factors for both 72 and 24, and correctly states that all prime factors of 24 are present in 72, making it divisible.

Real World Connection

In the Real World

This concept is fundamental in Cryptography, which is the science of secure communication. When you send money using UPI or your bank app, strong encryption algorithms are used to protect your data. These algorithms often rely on the properties of prime numbers and their factors to create codes that are extremely difficult to break, ensuring your transactions are safe.

Key Vocabulary

Key Terms

Divisibility: The ability of one number to be divided by another number without leaving a remainder. | Prime Factor: A factor of a number that is also a prime number. | Dividend: The number being divided. | Divisor: The number by which another number is divided. | Composite Number: A whole number that can be formed by multiplying two smaller positive integers.

What's Next

What to Learn Next

Great job understanding this! Next, you should explore the concepts of LCM (Least Common Multiple) and HCF (Highest Common Factor). They build directly on prime factorization and are super useful for solving problems involving fractions, time, and quantities.