Prime factorization, a foundational concept in mathematics, holds significant importance. It is a key tool for simplifying fractions, finding the greatest common divisor (GCD), and solving real-world problems. In this blog, we’ll delve into the importance of prime factorization and explain how to find the prime factors of 75 using the division method, a straightforward approach to factorization.
What Are Prime Factors?
Prime factors are the building blocks of a number. These are factors that are prime numbers — numbers greater than 1 that can only be divided by 1 and themselves. For example:
In this blog, we’ll focus on finding the prime factors of the number 75.
What is the Division Method?
The division method is a systematic and reliable process for finding a number’s prime factors. It involves dividing the number by the smallest possible prime numbers in ascending order until the result is 1. This ensures that all factors are prime.
What are the Factors of 75?
The factors of 75 are the numbers that can be paired through multiplication to produce the number 75. But, they are the numbers that divide 75 ultimately, leaving no remainder. Since 75 is a composite number, it has multiple factors aside from just 1 and itself. Therefore, the complete list of factors of 75 includes 1, 3, 5, 15, 25, and 75.
| Factors of 75 | |
|---|---|
| 1 | 3 |
| 5 | 15 |
| 25 | 75 |
Pair Factors of 75
To find the pair factors, multiply the two numbers in a pair to get the original number as 75; such numbers are as follows.
If 1 × 75 = 75, then (1, 75) is a pair factor of 75.
Similarly, let us find another pair.
3 × 25 = 75, (3, 25) is a pair factor of 75
5 × 15 = 75, (5, 15) is a pair factor of 75
25 × 3 = 75, (25, 3) is a pair factor of 75
15 × 5 = 75, (15, 5) is a pair factor of 75
Here,(3, 25) is the same as (25, 3) and (5, 15) is the same as (15, 5)
Therefore, the positive pair factors of 75 are (1, 75), (3, 25), and (5, 15).
To find the negative pair factors, then proceed with the following steps
If -1 × -75 = 75, then (-1,- 75) is a pair factor of 75
-3 × -25 = 75, (-3, -25) is a pair factor of 75
-5 × -15 = 75, (-5, -15) is a pair factor of 75
-25 × -3 = 75, (-25, -3) is a pair factor of 75
-15 × -5 = 75, (-15, -5) is a pair factor of 75
Here,(-3, -25) is the same as (-25, -3) and (-5, -15) is the same as (-15, -5)
Therefore, the negative pair factors of 75 are (-1, -75), (-3, -25), and (-5, – 15).
Step-by-Step Guide: Finding the Prime Factors of 75
Let’s break down the process step by step:
Step 1: Start with the Smallest Prime Number
- The smallest prime number is 2. Check if 75 is divisible by 2.
- Since 75 is an odd number, it’s not divisible by 2.
Step 2: Move to the Next Prime Number (3)
Divide 75 by 3: 75 ÷ 3 =25
Write this as: 75= 3×25
Step 3: Continue Dividing
- Now, focus on the quotient, 25. Check if it’s divisible by the next prime number: 25÷5=5
- Write this as: 75=3×5×5
Step 4: Finalize the Prime Factorization
- The quotient is now 5, which is itself a prime number. We stop dividing when the quotient becomes 1.
- Thus, the prime factorization of 75 is: 3×5×5
- Alternatively, we can express this as: 3× 52
Prime Factorization of 75
The number 75 is a composite, and it should have prime factors. Now let us know how to calculate the prime factors.
- Start by dividing 75 by the smallest prime number, which is 2. Since 75÷2 gives a fraction, 2 is not a factor of 75. Move to the next smallest prime number, 3, and try dividing again.
- 75 ÷ 3 = 25
- Now, if we divide 25 by 3, we will get a fractional number, which cannot be a factor.
- So, now proceed with the next prime numbers, i.e.
25 ÷ 5 = 5
5 ÷ 5 = 1
Finally, we received the number 1 at the end of the division process. So that we cannot proceed further. So, the prime factors of 75 are written as 3 × 5 × 5 or 3 x 52 , where 3 and 5 are the prime numbers.
Python Code for Prime Factorization
Here’s a simple implementation using Python:
def prime_factors(n): factors = [ ] # Check for number of 2s while n % 2 == 0: factors.append(2) n //= 2 # Check for odd factors from 3 onwards for i in range(3, int(n**0.5) + 1, 2): while n % i == 0: factors.append(i) n //= i # If n is still a prime number greater than 2 if n > 2: factors.append(n) return factors # Example print(prime_factors(75)) |
Output
[3, 5, 5] |
Examples
Example: 1
Find the common factors of 75 and 73.
Solution:
- Factors of 75: 1,3,5,15,25,75.
- Factors of 73: 1,73.
The only common factor is 1
Example 2:
Find the common factors of 75 and 76.
Solution:
- Factors of 75: 1,3,5,15,25,75
- Factors of 76: 1,2,4,19,38,76
The only common factor is 1.
Example 3:
Find the common factors of 75 and 150.
Solution:
- Factors of 75: 1,3,5,15,25,75.
- Factors of 150: 1,2,3,5,6,10,15,25,30,50,75,150.
The common factors are 1,3,5,15,25,75.
the prime factors of other numbers, like 60 or 100? Remember, practice makes perfect and it’s a great way to stay engaged with the topic!
FAQS
Q1: What are the prime factors of 75?
The prime factors of 75 are 3 and 5. Written as a product, it’s 3×52
Q2: Why isn’t 75 divisible by 2?
75 is an odd number, and only even numbers are divisible by 2.
Q3: Can I use other methods to find prime factors?
Yes! You can use a factor tree (as shown above) or continuous division. The division method is particularly useful because it’s systematic and works well for small numbers.
Q4: Why is prime factorization important?
Prime factorization has numerous applications:
Simplifying fractions
Finding the greatest common divisor (GCD) or least common multiple (LCM)
Cryptography and secure communications
Understanding the properties of numbers in number theory
Conclusion
Prime factorization is a simple yet powerful tool in mathematics. Using the division method, we found that the prime factors of 75 are 3×52 . This approach is systematic and reliable and helps us uncover the fundamental structure of numbers. This is how Moonpreneur teaches math.
Now that you’ve mastered the steps, why not try finding the prime factors of other numbers, like 60 or 100? Remember, practice makes perfect and it’s a great way to stay engaged with the topic!
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