What is the Sum of Divisors of a Number?

S3-SA4-0236

Grade Level:

Class 8

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

Definition

What is it?

The sum of divisors of a number is the total you get when you add up all its positive factors (divisors). A divisor is a number that divides another number exactly, leaving no remainder.

Simple Example

Quick Example

Imagine you have 6 mangoes. You want to divide them equally among your friends. You can give 1 mango to 6 friends, 2 mangoes to 3 friends, 3 mangoes to 2 friends, or 6 mangoes to 1 friend. The divisors are 1, 2, 3, and 6. Their sum is 1 + 2 + 3 + 6 = 12.

Worked Example

Step-by-Step

Let's find the sum of divisors for the number 12.

Step 1: Find all the positive divisors of 12. These are the numbers that divide 12 exactly without leaving a remainder.
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Step 2: Start checking from 1. Is 1 a divisor of 12? Yes (12 / 1 = 12).
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Step 3: Is 2 a divisor of 12? Yes (12 / 2 = 6).
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Step 4: Is 3 a divisor of 12? Yes (12 / 3 = 4).
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Step 5: Is 4 a divisor of 12? Yes (12 / 4 = 3).
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Step 6: Is 5 a divisor of 12? No (12 / 5 leaves a remainder).
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Step 7: Is 6 a divisor of 12? Yes (12 / 6 = 2). We can stop here because the next divisor will be 12 itself.
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Step 8: The divisors of 12 are 1, 2, 3, 4, 6, and 12. Now, add them up: 1 + 2 + 3 + 4 + 6 + 12 = 28.

Answer: The sum of divisors of 12 is 28.

Why It Matters

Understanding the sum of divisors helps in advanced math problems like number theory. It's used in computer science for algorithms and in cryptography to secure online transactions, protecting your mobile banking and WhatsApp chats. Even data scientists use properties of numbers to analyze complex information.

Common Mistakes

MISTAKE: Forgetting to include 1 and the number itself as divisors. | CORRECTION: Always remember that 1 and the number itself are always divisors of any positive integer.

MISTAKE: Listing only prime factors instead of all factors. For example, for 10, listing only 2 and 5. | CORRECTION: Make sure to list all factors (1, 2, 5, 10 for 10), not just the prime ones.

MISTAKE: Missing some divisors, especially for larger numbers. | CORRECTION: To avoid missing divisors, list them in pairs. For example, for 20: (1, 20), (2, 10), (4, 5). This ensures you find all of them.

Practice Questions

Try It Yourself

QUESTION: What is the sum of divisors of the number 7? | ANSWER: 8

QUESTION: Find the sum of divisors for the number 15. | ANSWER: 24

QUESTION: A shopkeeper wants to arrange 24 ladoos in equal rows. What is the sum of all possible numbers of ladoos that can be in one row (i.e., the sum of divisors of 24)? | ANSWER: 60

MCQ

Quick Quiz

What is the sum of divisors of the number 10?

The Correct Answer Is:

The divisors of 10 are 1, 2, 5, and 10. Adding them together gives 1 + 2 + 5 + 10 = 18. Options A, B, and D are incorrect because they don't include all divisors or sum them incorrectly.

Real World Connection

In the Real World

In computer science, especially in cryptography, numbers with specific sums of divisors (like 'perfect numbers') are studied. These properties help create strong encryption algorithms that protect your data when you use apps like Google Pay or send messages on Signal, ensuring your information is safe from hackers.

Key Vocabulary

Key Terms

DIVISOR: A number that divides another number exactly, leaving no remainder. | FACTOR: Another name for a divisor. | PRIME NUMBER: A number greater than 1 that has only two divisors: 1 and itself. | COMPOSITE NUMBER: A number greater than 1 that has more than two divisors.

What's Next

What to Learn Next

Next, you can explore 'Perfect Numbers' and 'Abundant Numbers'. These concepts directly use the sum of divisors to classify numbers, showing how this foundational idea leads to fascinating discoveries in number theory.