factors of 252 in pairs

Daniel Parker

2 min read

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25-02-2025

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Finding the factors of a number and expressing them in pairs is a fundamental concept in number theory. This article delves into the process of identifying all factor pairs of 252, explaining the method and providing a clear, step-by-step approach. We'll also explore some related concepts to build a strong understanding.

Understanding Factors

Before we dive into the factor pairs of 252, let's clarify what a factor is. A factor of a number is a 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 because each of these numbers divides perfectly into 12.

Prime Factorization: The Foundation

Prime factorization is a crucial step in finding all the factors of a number. Prime factorization means expressing a number as a product of its prime factors. A prime number is a whole number greater than 1 that has only two divisors: 1 and itself. Let's 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

This means 252 can be written as 2 squared times 3 squared times 7.

Listing Factor Pairs of 252

Now that we have the prime factorization, we can systematically find all the factor pairs. We do this by considering all possible combinations of the prime factors.

Here's how we can approach it:

1. Start with the smallest factors: The smallest factor is always 1. Its pair is 252 (1 x 252 = 252).

2. Consider combinations of prime factors:

• 2: Its pair is 126 (2 x 126 = 252).
• 3: Its pair is 84 (3 x 84 = 252).
• 4 (2²): Its pair is 63 (4 x 63 = 252).
• 6 (2 x 3): Its pair is 42 (6 x 42 = 252).
• 7: Its pair is 36 (7 x 36 = 252).
• 9 (3²): Its pair is 28 (9 x 28 = 252).
• 12 (2² x 3): Its pair is 21 (12 x 21 = 252).
• 14 (2 x 7): Its pair is 18 (14 x 18 = 252).
• 18 (2 x 3²): Its pair is 14 (as calculated above).
• 21 (3 x 7): Its pair is 12 (as calculated above).
• 28 (2² x 7): Its pair is 9 (as calculated above).
• 36 (2² x 3²): Its pair is 7 (as calculated above).
• 42 (2 x 3 x 7): Its pair is 6 (as calculated above).
• 63 (3² x 7): Its pair is 4 (as calculated above).
• 84 (2² x 3 x 7): Its pair is 3 (as calculated above).
• 126 (2 x 3² x 7): Its pair is 2 (as calculated above).

3. The largest factor: The largest factor is 252 itself. Its pair is 1 (as calculated above).

Therefore, the factor pairs of 252 are: (1, 252), (2, 126), (3, 84), (4, 63), (6, 42), (7, 36), (9, 28), (12, 21), (14, 18).

Visualizing Factor Pairs

A visual representation can be helpful. We could create a factor tree to show the prime factorization, and then use that to systematically identify the pairs. However, a simple list, as presented above, is perfectly sufficient for understanding and solving this type of problem.

Conclusion

Finding the factor pairs of 252, or any number, involves understanding prime factorization and systematically exploring combinations of the prime factors. By following these steps, you can confidently determine all the factor pairs of any given number. Remember, this process is essential in various mathematical concepts and problem-solving situations.