Prime Factors: Rules, Methods and Solved Examples
Prime Factors are the prime numbers that divide a given integer exactly. This page explains prime factorization using repeated division and factor trees, divisibility-based shortcuts, unique factorization, and applications in finding HCF, LCM, and the number of factors. Worked examples show how to break composite numbers into their prime components.
What Are Prime Factors?
For example, 60 = 2 × 2 × 3 × 5 = 2² × 3 × 5, so the prime factors of 60 are 2, 3, and 5. The factorization is unique apart from the order of the prime factors. The number 1 has no prime factors because 1 is neither prime nor composite.
Prime Factors Formula & Tricks
Important Formulas
Here, p₁, p₂, ..., pₖ are distinct prime numbers and a₁, a₂, ..., aₖ are positive integers.
If n = p₁ᵃ¹ × p₂ᵃ² × ... × pₖᵃᵏ, each exponent can be selected from 0 through its given value.
For each prime common to the numbers, choose the minimum exponent.
For each prime appearing in at least one factorization, choose the maximum exponent.
Quick Tricks
To factor a number, try division by 2, 3, 5, 7, 11 and other primes instead of every integer. After removing a prime factor repeatedly, continue with the remaining quotient.
If the remaining number is greater than 1 and no prime divisor up to its square root divides it, the remaining number is prime. After removing a factor p, test the quotient rather than the original number.
A last digit of 0, 2, 4, 6, or 8 indicates divisibility by 2; a digit sum divisible by 3 indicates divisibility by 3; a last digit of 0 or 5 indicates divisibility by 5.
Prime Factors Concepts
Repeated Division Method
Write the prime divisors obtained at each step and multiply them to form the factorization. Divide by the same prime again whenever it still divides the quotient.
Factor Tree Method
The order of splitting can differ, but the final multiset of prime factors is the same. Combine repeated prime factors using exponents.
Prime Factorization and Uniqueness
A prime number has only itself as its prime factor. A composite number has at least two prime factors when counted with repetition. For example, 72 = 2³ × 3²; its distinct prime factors are 2 and 3, while its prime factors counted with repetition are 2, 2, 2, 3, 3.
Applications of Exponents in Factorization
For 72 = 2³ × 3², the number of positive factors is (3 + 1)(2 + 1) = 12. For 72 and 90 = 2 × 3² × 5, HCF = 2¹ × 3² = 18 and LCM = 2³ × 3² × 5 = 360.
Prime Factors Video Lessons
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Multiples and Factors in Number System
Understand the meaning of multiples and factors, learn how they are related, and identify them using simple examples in the Number System.
Practice Prime Factors Questions
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Prime Factors Revision Points
Use these rules to revise prime factorization and its common applications.
- A prime number has exactly two positive factors: 1 and itself.
- The number 1 has no prime factors.
- Every integer greater than 1 has a unique prime factorization, apart from order.
- Use repeated division or a factor tree to obtain prime factors.
- For factor testing, only prime divisors up to the square root of the remaining number are required.
- If n = p₁ᵃ¹ × p₂ᵃ² × ... × pₖᵃᵏ, its number of positive factors is (a₁ + 1)(a₂ + 1)...(aₖ + 1).
- For HCF, use common primes with minimum exponents; for LCM, use all primes with maximum exponents.
Prime Factors FAQs
What is the prime factorization of 144?
144 = 2 × 2 × 2 × 2 × 3 × 3 = 2⁴ × 3².
How are distinct prime factors different from prime factors counted with repetition?
For 60 = 2² × 3 × 5, the distinct prime factors are 2, 3, and 5. Counting repetition gives 2, 2, 3, and 5.
How many positive factors does 360 have?
Since 360 = 2³ × 3² × 5, the number of factors is (3 + 1)(2 + 1)(1 + 1) = 24.
What is the prime factorization of a prime number such as 29?
The prime factorization of 29 is 29 itself because 29 has no prime divisor other than 1 and 29.
How do prime factors help find the HCF of 48 and 180?
48 = 2⁴ × 3 and 180 = 2² × 3² × 5. Taking common primes with smaller exponents gives HCF = 2² × 3 = 12.
How do prime factors help find the LCM of 18 and 45?
18 = 2 × 3² and 45 = 3² × 5. Taking every prime with its greater exponent gives LCM = 2 × 3² × 5 = 90.