Special Numbers and Their Properties

 

1. Perfect Numbers

Definition

A perfect number is a positive integer that is equal to the sum of its proper divisors (excluding itself).

Examples of Perfect Numbers

1. 6 → Proper divisors: 1, 2, 3
   • 1 + 2 + 3 = 6 ✅ (Perfect Number)
2. 28 → Proper divisors: 1, 2, 4, 7, 14
   • 1 + 2 + 4 + 7 + 14 = 28 ✅ (Perfect Number)
3. 496 → Proper divisors: 1, 2, 4, 8, 16, 31, 62, 124, 248
   • 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248 = 496 ✅ (Perfect Number)

MCQs on Perfect Numbers

1. Which of the following is a perfect number?

a) 10

b) 15

c) 28

d) 30

✅ Answer: c) 28

2. The smallest perfect number is:

a) 5

b) 6

c) 7

d) 9

✅ Answer: b) 6

2. Armstrong Numbers

Definition

An Armstrong number (also called a narcissistic number) is a number that is equal to the sum of its digits raised to the power of the number of digits.

Examples of Armstrong Numbers

1. 153 → 3-digit number
   • 1³ + 5³ + 3³ = 1 + 125 + 27 = 153 ✅ (Armstrong Number)
2. 9474 → 4-digit number
   • 9⁴ + 4⁴ + 7⁴ + 4⁴ = 6561 + 256 + 2401 + 256 = 9474 ✅ (Armstrong Number)

MCQs on Armstrong Numbers

3. Which of the following is an Armstrong number?

a) 123

b) 370

c) 456

d) 700

✅ Answer: b) 370

4. 9474 is an Armstrong number. How many digits does it have?

a) 3

b) 4

c) 5

d) 6

✅ Answer: b) 4

3. Fibonacci Series

Definition

The Fibonacci series is a sequence of numbers where each number is the sum of the two preceding numbers, starting from 0 and 1.

Formula

F(n)=F(n−1)+F(n−2)F(n)=F(n−1)+F(n−2)

Where:

• F(0) = 0
• F(1) = 1

First Few Fibonacci Numbers

0,1,1,2,3,5,8,13,21,34,...0,1,1,2,3,5,8,13,21,34,...

Example Calculation

• F(5) = F(4) + F(3) = 3 + 2 = 5
• F(6) = F(5) + F(4) = 5 + 3 = 8

MCQs on Fibonacci Series

5. What is the 7th number in the Fibonacci series?

a) 8

b) 13

c) 21

d) 5

✅ Answer: b) 13

6. What are the first two numbers in the Fibonacci series?

a) 1, 2

b) 2, 3

c) 0, 1

d) 1, 1

✅ Answer: c) 0, 1

4. Square and Cube Numbers

Square Numbers

A square number (also called a perfect square) is a number that is the square of an integer.

n2=n×nn2=n×n

Examples

• 1² = 1
• 2² = 4
• 3² = 9
• 4² = 16
• 5² = 25

Cube Numbers

A cube number (also called a perfect cube) is a number that is the cube of an integer.

n3=n×n×nn3=n×n×n

Examples

• 1³ = 1
• 2³ = 8
• 3³ = 27
• 4³ = 64
• 5³ = 125

MCQs on Square and Cube Numbers

7. What is the square of 12?

a) 122

b) 120

c) 144

d) 128

✅ Answer: c) 144

8. What is the cube of 4?

a) 12

b) 16

c) 64

d) 24

✅ Answer: c) 64