What are Natural Numbers?

Natural numbers are the set of positive integers starting from 1 and extending infinitely. They are denoted by N = {1, 2, 3, 4, ...}.

Define Whole Numbers.

Whole numbers include all natural numbers along with zero. 
They are represented as W = {0, 1, 2, 3, ...}.

What is the difference between rational and irrational numbers? 

(Hint: Think about their representation)

Rational numbers can be expressed as a fraction of two integers (e.g., 1/2, 3), while irrational numbers cannot be expressed as fractions and have non-terminating, non-repeating decimal expansions (e.g., √2, π).

What is an Integer?

An integer is any whole number, positive or negative, including zero. The set of integers is denoted by Z = {..., -3, -2, -1, 0, 1, 2, 3, ...}.

Define Prime Numbers.

A Prime Number is a natural number greater than 1 that has no divisors other than 1 and itself. Examples include 2, 3, 5, 7, and 11.

What are Composite Numbers?

Composite numbers are natural numbers greater than 1 that have more than two distinct positive divisors. 

For Example: 4, 6, 8, and 9 are composite numbers.

How do you check if a number is prime? 

(Hint: Consider factors up to the square root)

To check if a number N is prime, find the square root of N, round it up to the nearest whole number, and test for divisibility by all prime numbers less than or equal to that number.

Explain the Absolute Value of a number.

The absolute value of a number is its distance from zero on the number line, regardless of direction. 
It is denoted as |a|, where |a| = a if a ≥ 0, and |a| = -a if a < 0.

What is the least common multiple (LCM)? 

(Hint: Think about multiples)

The least common multiple (LCM) of two or more integers is the smallest positive integer that is divisible by each of the integers. 
For Example, 
LCM of 4 and 5 is 20.

How do you find the Highest Common Factor (HCF)?

The highest common factor (HCF) is found by determining the largest number that divides two or more numbers without leaving a remainder. 

It can be calculated using prime factorization.

What is a Perfect Number? Give an example.

A perfect number is a positive integer that is equal to the sum of its proper divisors, excluding itself. The smallest perfect number is 6, since 1 + 2 + 3 = 6.

What does it mean for two numbers to be co-prime? 

(Hint: Consider their factors)

Two numbers are co-prime if their greatest common divisor (GCD) is 1, meaning they have no prime factors in common. Example: 8 and 9 are co-prime.

Explain the concept of Divisibility.

A number A is divisible by another number B if A can be divided by B without leaving a remainder. 
For Example, 
10 is divisible by 5 since
 10 ÷ 5 = 2.

What are Even and Odd numbers?

Even numbers are integers that are divisible by 2 (e.g., -4, 0, 2, 6), while odd numbers are integers that are not divisible by 2 (e.g., -3, 1, 5).

What is the prime factorization of a number? 

(Hint: Consider breaking it down into primes)

Prime factorization is expressing a number as the product of its prime factors. 
For Example, 
the prime factorization of 28 is 2² * 7.

What is the relationship between Prime and Composite Numbers?

Every composite number can be factored into prime numbers, while prime numbers cannot be factored any further except by 1 and themselves.

How is the number of trailing zeroes in a factorial determined? Hint: Think about the factors of 10.

The number of trailing zeros in n! is determined by the number of times 5 is a factor in the numbers from 1 to n. This is because pairs of 2 and 5 form 10.

What defines a rational number? Provide an example.

A rational number is any number that can be expressed as a fraction p/q, where p and q are integers and q ≠ 0. 
For Example, 3/4 and -2/5 are rational.

What is the decimal representation of a fraction? 

(Hint: Think about the conversion process)

A fraction can be converted to a decimal by dividing the numerator by the denominator. For example, 1/4 = 0.25.

How do you convert a non-terminating decimal into a fraction? 

Let x = 0.333... (repeating). Then, multiply both sides by 10: 10x = 3.333... . 
Subtract: 
10x - x = 3. 
Thus, 9x = 3, 
leading to x = 1/3.

What is the significance of the number zero in the number system?

Zero is a unique integer that represents a null value and serves as an identity element for addition. It is neither positive nor negative.

What is a mixed number? 

(Hint: Think about combining whole and fractional parts)

A mixed number is a combination of a whole number and a proper fraction, such as , which represents 2 + 1/3.

Define a common fraction.

A common fraction is a fraction whose denominator is not a power of 10, such as 3/4 or 5/8.

What does the term 'decimal fraction' refer to?

A decimal fraction is a fraction where the denominator is a power of 10, such as 0.5 (which is 5/10) or 0.75 (which is 75/100).

What are the rules for divisibility by 2? 

(Hint: Consider the last digit of a number)

A number is divisible by 2 if its last digit is even, i.e., 0, 2, 4, 6, or 8. For example, 24 and 58 are divisible by 2.