What is an Amicable Number?

S3-SA4-0228

Grade Level:

Class 8

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

Definition

What is it?

Amicable numbers are two different positive whole numbers where the sum of the proper divisors of each number (divisors excluding the number itself) equals the other number. Think of them as a 'friendly pair' where each number 'gives back' to the other through its divisors.

Simple Example

Quick Example

Imagine two friends, Rohan and Priya. Rohan has a secret list of his favourite numbers (proper divisors). If he adds them up, the total is exactly Priya's favourite number. Priya also has her own secret list, and when she adds them up, the total is exactly Rohan's favourite number. That's how 220 and 284 are amicable numbers.

Worked Example

Step-by-Step

Let's check if 220 and 284 are an amicable pair:

1. Find the proper divisors of 220.
Proper divisors of 220 are: 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110.

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2. Sum the proper divisors of 220.
Sum = 1 + 2 + 4 + 5 + 10 + 11 + 20 + 22 + 44 + 55 + 110 = 284.

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3. Find the proper divisors of 284.
Proper divisors of 284 are: 1, 2, 4, 71, 142.

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4. Sum the proper divisors of 284.
Sum = 1 + 2 + 4 + 71 + 142 = 220.

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5. Compare the sums.
The sum of proper divisors of 220 is 284, and the sum of proper divisors of 284 is 220.

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ANSWER: Since the sum of proper divisors of 220 is 284, and the sum of proper divisors of 284 is 220, 220 and 284 are indeed an amicable pair.

Why It Matters

Amicable numbers are more than just a math curiosity; they help us understand relationships between numbers, which is fundamental in computer science and cryptography. In fields like data science, understanding number theory helps in designing efficient algorithms and secure systems, essential for protecting your online banking or UPI transactions.

Common Mistakes

MISTAKE: Including the number itself when summing divisors. For example, for 220, adding 220 to its divisors. | CORRECTION: Remember 'proper divisors' means all positive divisors EXCEPT the number itself. Always exclude the number you are finding divisors for.

MISTAKE: Missing some proper divisors, leading to an incorrect sum. For example, forgetting 11 is a divisor of 220. | CORRECTION: Be systematic. Start checking from 1 upwards, testing divisibility for each number until you reach half of the original number. Don't forget 1 is always a divisor.

MISTAKE: Confusing amicable numbers with perfect numbers or abundant/deficient numbers. | CORRECTION: Amicable numbers are a PAIR where each sum equals the OTHER number. A perfect number is one where the sum of its proper divisors equals ITSELF. Abundant/deficient numbers relate to whether the sum is greater or less than the number itself.

Practice Questions

Try It Yourself

QUESTION: Are 30 and 40 an amicable pair? | ANSWER: No, they are not. Proper divisors of 30: 1, 2, 3, 5, 6, 10, 15 (sum = 42). Proper divisors of 40: 1, 2, 4, 5, 8, 10, 20 (sum = 50).

QUESTION: The sum of proper divisors of 1184 is 1210. What number must be the sum of proper divisors of 1210 for them to be an amicable pair? | ANSWER: For 1184 and 1210 to be an amicable pair, the sum of proper divisors of 1210 must be 1184.

QUESTION: Find the sum of the proper divisors for the number 6. Is it an amicable number with any other number? | ANSWER: Proper divisors of 6 are 1, 2, 3. Sum = 1 + 2 + 3 = 6. Since the sum of its proper divisors is 6 (the number itself), 6 is a perfect number, not part of an amicable pair.

MCQ

Quick Quiz

Which of the following statements is true about amicable numbers?

They are single numbers whose proper divisors sum to themselves.

They are pairs of numbers where each number's proper divisors sum to the other number.

They are numbers that have no proper divisors.

They are numbers that are only divisible by 1 and themselves.

The Correct Answer Is:

Option B correctly defines amicable numbers as a pair where the sum of proper divisors of one equals the other. Option A describes perfect numbers, while C and D describe properties not related to amicable numbers.

Real World Connection

In the Real World

While not directly used in everyday apps like WhatsApp, the underlying number theory concepts, including properties of divisors, are crucial in cryptography. For example, securing your online transactions or protecting your personal data on platforms like Aadhaar uses complex mathematical ideas about prime numbers and divisibility, which are closely related to understanding numbers like amicable pairs.

Key Vocabulary

Key Terms

PROPER DIVISOR: A positive divisor of a number, excluding the number itself. | SUM: The result of adding two or more numbers. | PRIME NUMBER: A whole number greater than 1 that has exactly two proper divisors: 1 and itself. | COMPOSITE NUMBER: A whole number greater than 1 that has more than two proper divisors. | PERFECT NUMBER: A number where the sum of its proper divisors equals the number itself.

What's Next

What to Learn Next

Great job learning about amicable numbers! Next, explore 'Perfect Numbers' and 'Abundant and Deficient Numbers'. These concepts build on proper divisors and will help you classify numbers based on the sum of their divisors, giving you an even deeper understanding of number theory.