What is a Friendly Number?

S3-SA4-0307

Grade Level:

Class 6

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

Definition

What is it?

Friendly Numbers are two different numbers that share the same 'abundancy index'. The abundancy index is found by dividing the sum of all a number's factors (including the number itself) by the number itself. If two different numbers have the exact same abundancy index, they are called a 'friendly pair'.

Simple Example

Quick Example

Imagine two friends, Rohan and Priya, each have a box of laddoos. Rohan has 6 laddoos and Priya has 28 laddoos. If we calculate a special 'sweetness ratio' for each box based on how many laddoos are there and how many ways you can share them, and both Rohan's and Priya's boxes have the exact same 'sweetness ratio', then their numbers (6 and 28) would be a friendly pair!

Worked Example

Step-by-Step

Let's check if 30 and 140 are a friendly pair.
---STEP 1: Find factors of 30. Factors of 30 are 1, 2, 3, 5, 6, 10, 15, 30.
---STEP 2: Sum the factors of 30. 1 + 2 + 3 + 5 + 6 + 10 + 15 + 30 = 72.
---STEP 3: Calculate the abundancy index for 30. Abundancy Index = Sum of factors / Number = 72 / 30 = 2.4.
---STEP 4: Find factors of 140. Factors of 140 are 1, 2, 4, 5, 7, 10, 14, 20, 28, 35, 70, 140.
---STEP 5: Sum the factors of 140. 1 + 2 + 4 + 5 + 7 + 10 + 14 + 20 + 28 + 35 + 70 + 140 = 336.
---STEP 6: Calculate the abundancy index for 140. Abundancy Index = Sum of factors / Number = 336 / 140 = 2.4.
---STEP 7: Compare the abundancy indices. Both 30 and 140 have an abundancy index of 2.4.
---ANSWER: Yes, 30 and 140 are a friendly pair because they have the same abundancy index (2.4).

Why It Matters

Understanding number properties like Friendly Numbers helps build a strong foundation for advanced mathematics. In Computer Science, these concepts are used in algorithms for data analysis and cryptography. Engineers and data scientists use number theory to design efficient systems and secure information.

Common Mistakes

MISTAKE: Forgetting to include the number itself when summing its factors. | CORRECTION: Always include the number itself in the list of factors when calculating the sum for the abundancy index.

MISTAKE: Thinking that any two numbers with common factors are friendly. | CORRECTION: Friendly numbers require the *same abundancy index*, not just common factors. The abundancy index is a specific ratio.

MISTAKE: Calling a number friendly to itself. | CORRECTION: Friendly Numbers are always a *pair* of *different* numbers. A number cannot be friendly with itself.

Practice Questions

Try It Yourself

QUESTION: Is 6 a friendly number with 12? (Hint: Factors of 6 are 1, 2, 3, 6. Factors of 12 are 1, 2, 3, 4, 6, 12.) | ANSWER: No, 6 and 12 are not a friendly pair. Abundancy index of 6 = (1+2+3+6)/6 = 12/6 = 2. Abundancy index of 12 = (1+2+3+4+6+12)/12 = 28/12 = 2.33 (approx). Since 2 is not equal to 2.33, they are not friendly.

QUESTION: Find the abundancy index for the number 20. | ANSWER: Factors of 20 are 1, 2, 4, 5, 10, 20. Sum of factors = 1+2+4+5+10+20 = 42. Abundancy Index = 42/20 = 2.1.

QUESTION: The abundancy index for 120 is 3.6. If a number X has factors 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120, and its sum of factors is 360, is X a friendly number with 120? | ANSWER: Yes, X is a friendly number with 120. The problem states the sum of factors for X is 360 and the factors are 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120. This means X must be 120. However, friendly numbers are *different* numbers. But if we assume X is a *different* number whose sum of factors is 360 and the number itself is 100 (for example, if X was 100 and its factors summed to 360), then its abundancy index would be 360/100 = 3.6. Since 120's abundancy index is also 3.6, if X was a different number with an abundancy index of 3.6, then they would be friendly. The question implies X is a number different from 120 but shares the same abundancy index. So, if X is a number whose sum of factors is 360 and the number itself is 100, then its abundancy index is 3.6, making it friendly with 120. The question is a bit tricky as the list of factors provided is for 120 itself. If X is a *different* number that happens to have an abundancy index of 3.6, then it would be friendly with 120. Let's assume X is a different number, say 100, that has an abundancy index of 3.6. Then yes, they would be friendly. (A more precise question would give X directly.)

MCQ

Quick Quiz

Which of these is the correct way to calculate the abundancy index of a number?

Sum of factors (excluding the number itself) / Number

Sum of factors (including the number itself) / Number

Number / Sum of factors (including the number itself)

Product of factors / Number

The Correct Answer Is:

The abundancy index is defined as the sum of all factors of a number, *including* the number itself, divided by the number itself. Option B correctly states this definition.

Real World Connection

In the Real World

While Friendly Numbers themselves aren't used daily by a delivery driver or a chaiwala, the underlying concepts of factorisation and ratios are vital. For example, in computer security, complex algorithms based on prime factors (a type of factor) are used to encrypt your UPI transactions and protect your mobile data, ensuring your money and information are safe.

Key Vocabulary

Key Terms

FACTOR: A number that divides another number exactly, without leaving a remainder. | SUM OF FACTORS: The total you get when you add up all the factors of a number. | ABUNDANCY INDEX: A special ratio found by dividing the sum of a number's factors by the number itself. | FRIENDLY PAIR: Two different numbers that have the exact same abundancy index.

What's Next

What to Learn Next

Great job understanding Friendly Numbers! Next, you can explore 'Perfect Numbers', which are special numbers where the sum of their *proper* factors (factors excluding the number itself) equals the number itself. This builds on your understanding of factors and sums!