Understanding the Problem
We are given a number n
, and our task is to find its prime factorization — that is, express it as a product of prime numbers.
To do this efficiently, especially when we have to factorize many numbers, we can use a technique based on the Sieve of Eratosthenes. Instead of just marking prime numbers, we compute the Smallest Prime Factor (SPF) for every number up to a certain limit.
Once we have this SPF array, we can use it to find the prime factors of any number ≤ limit quickly by continuously dividing the number by its SPF until it becomes 1.
Step-by-Step Solution with Example
step 1: Precompute the Smallest Prime Factor (SPF)
We initialize an array spf[]
such that spf[i] = i
initially.
We iterate from 2 to limit
and if i
is a prime (i.e., spf[i] == i
), then for every multiple j
of i
, we set spf[j] = i
only if spf[j] == j
. This ensures that each number gets its smallest prime divisor.
step 2: Use SPF to Factorize the Number
Let’s say we want to factorize n = 60
. We use the spf[]
array and do the following:
- Start with
n = 60
- spf[60] = 2 ⇒ store 2, then divide: n = 60 / 2 = 30
- spf[30] = 2 ⇒ store 2, then divide: n = 30 / 2 = 15
- spf[15] = 3 ⇒ store 3, then divide: n = 15 / 3 = 5
- spf[5] = 5 ⇒ store 5, then divide: n = 5 / 5 = 1
We stop when n
becomes 1. So, the prime factorization of 60 is: 2, 2, 3, 5
step 3: Output the Result
We now have a list of prime factors that can be printed or returned as needed.
Edge Cases
- n = 1: No prime factors, since 1 is neither prime nor composite. Return an empty list.
- n is a prime: For example, n = 13 → spf[13] = 13 → 13 is its only prime factor.
- Repeated prime factors: Like 2 × 2 × 2 × 2 for n = 16. The algorithm handles this naturally as we keep dividing by the SPF until n becomes 1.
- Multiple queries: This approach is especially useful if we need to factorize multiple numbers, as SPF is computed once and reused.
Final Thoughts
This method is highly efficient for prime factorization when dealing with many queries. The preprocessing with the sieve takes O(n log log n), and each factorization takes O(log n) time due to repeated divisions by primes.
By understanding and visualizing each step, especially through the example of n = 60, beginners can grasp how the sieve helps break down numbers into primes quickly and systematically.
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