CLASS 10 MATHEMATICS
Chapter 1: Real Numbers • Revision Notes
1. The Fundamental Theorem of Arithmetic
Every composite number can be expressed (factorized) as a product of primes, and this
factorization is unique, apart from the order in which the prime factors occur.
• Composite Number: A number that has more than two factors (e.g., 4, 6, 8).
• Prime Number: A number that has exactly two factors: 1 and itself (e.g., 2, 3, 5, 7, 11).
Example: 120 = 2 × 2 × 2 × 3 × 5 = 2³ × 3 × 5
2. HCF and LCM
HCF (Highest Common Factor): Product of the smallest power of each common prime factor.
LCM (Lowest Common Multiple): Product of the greatest power of each prime factor involved.
Crucial Relationship Formula:
HCF(a, b) × LCM(a, b) = a × b
Note: This relationship holds true only for two positive integers.
3. Irrational Numbers
A number is irrational if it cannot be expressed in the form p/q, where p and q are integers and q
≠ 0.
• Theorem: If p is a prime number and p divides a², then p also divides a (where a is a positive
integer).
• Method of Contradiction: Used to prove irrationality (e.g., proving √5 is irrational).
Chapter 1: Real Numbers • Revision Notes
1. The Fundamental Theorem of Arithmetic
Every composite number can be expressed (factorized) as a product of primes, and this
factorization is unique, apart from the order in which the prime factors occur.
• Composite Number: A number that has more than two factors (e.g., 4, 6, 8).
• Prime Number: A number that has exactly two factors: 1 and itself (e.g., 2, 3, 5, 7, 11).
Example: 120 = 2 × 2 × 2 × 3 × 5 = 2³ × 3 × 5
2. HCF and LCM
HCF (Highest Common Factor): Product of the smallest power of each common prime factor.
LCM (Lowest Common Multiple): Product of the greatest power of each prime factor involved.
Crucial Relationship Formula:
HCF(a, b) × LCM(a, b) = a × b
Note: This relationship holds true only for two positive integers.
3. Irrational Numbers
A number is irrational if it cannot be expressed in the form p/q, where p and q are integers and q
≠ 0.
• Theorem: If p is a prime number and p divides a², then p also divides a (where a is a positive
integer).
• Method of Contradiction: Used to prove irrationality (e.g., proving √5 is irrational).
