factor pairs of 252

Charlotte

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27-02-2025

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Finding all the factor pairs of a number like 252 might seem daunting, but with a systematic approach, it becomes much easier. This article will guide you through the process, explaining the concept and providing different methods to discover all the factor pairs of 252. Understanding factors and factor pairs is fundamental in mathematics, particularly in algebra and number theory.

What are Factors and Factor Pairs?

A factor of a number is any whole number that divides evenly into that number without leaving a remainder. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12.

A factor pair consists of two factors that, when multiplied together, equal the original number. For example, (1, 12), (2, 6), and (3, 4) are all factor pairs of 12.

Method 1: Prime Factorization

The most efficient way to find all factor pairs is to begin with prime factorization. Prime factorization breaks a number down into its prime factors (numbers only divisible by 1 and themselves).

1. Find the prime factorization of 252:

252 = 2 x 126 = 2 x 2 x 63 = 2 x 2 x 3 x 21 = 2 x 2 x 3 x 3 x 7 = 2² x 3² x 7

2. Create factor pairs:

Now that we have the prime factorization (2² x 3² x 7), we can systematically create all possible factor pairs. Let's consider the exponents:

• 2: We can have 2⁰, 2¹, or 2². (1, 2, 4)
• 3: We can have 3⁰, 3¹, or 3². (1, 3, 9)
• 7: We can have 7⁰ or 7¹. (1, 7)

To get all the factors, we combine these possibilities:

1, 2, 3, 4, 6, 7, 9, 12, 14, 18, 21, 28, 36, 42, 63, 126, 252

3. List the factor pairs:

Now, we pair the factors to find the factor pairs of 252:

• (1, 252)
• (2, 126)
• (3, 84)
• (4, 63)
• (6, 42)
• (7, 36)
• (9, 28)
• (12, 21)
• (14, 18)

Method 2: Systematic Listing

This method is less efficient for larger numbers but can be helpful for smaller numbers or as a way to verify your results. You start by listing factors sequentially, checking if they divide 252 evenly.

1. Start with 1: 1 is always a factor. Its pair is 252 (252/1 = 252).

2. Try subsequent numbers: 2 divides 252 evenly (252/2 = 126), giving us the pair (2, 126). Continue this process with 3, 4, and so on.

3. Stop when you reach the square root: You only need to test numbers up to the square root of 252 (approximately 15.87). Once you exceed that, you'll start repeating pairs in reverse order.

Why is Understanding Factor Pairs Important?

Understanding factors and factor pairs is crucial for various mathematical concepts:

• Simplifying Fractions: Finding the greatest common factor (GCF) relies on identifying factors.
• Solving Equations: Factoring quadratic equations uses knowledge of factor pairs.
• Number Theory: Factorization is fundamental in many number theory problems.

This comprehensive guide shows different methods to determine all factor pairs of 252. Remember, prime factorization offers the most efficient approach for larger numbers. Mastering this skill enhances your mathematical understanding and problem-solving abilities.