Which Answer Choice Contains All The Factors Of 10

Which Answer Choice Contains All the Factors of 10? A Deep Dive into Factorization

Finding all the factors of a number might seem like a simple mathematical task, but understanding the underlying principles can unlock a deeper appreciation of number theory. This article delves into the question: Which answer choice contains all the factors of 10? We'll explore the concept of factors, prime factorization, and how to systematically identify all factors of a given number. Beyond simply answering the question, we aim to equip you with the tools to tackle similar problems with confidence.

Understanding Factors

A factor of a number is a whole number that divides evenly into that number without leaving a remainder. For example, the factors of 6 are 1, 2, 3, and 6 because each of these numbers divides evenly into 6. It's important to note that both 1 and the number itself are always factors.

Let's break down how to find factors systematically:

Method 1: Trial Division

The simplest approach is trial division. We systematically test each whole number, starting from 1, to see if it divides evenly into our target number. For small numbers, this is perfectly efficient. Let's illustrate with the number 12:

1. 1: 12 ÷ 1 = 12 (remainder 0) - 1 is a factor
2. 2: 12 ÷ 2 = 6 (remainder 0) - 2 is a factor
3. 3: 12 ÷ 3 = 4 (remainder 0) - 3 is a factor
4. 4: 12 ÷ 4 = 3 (remainder 0) - 4 is a factor
5. 5: 12 ÷ 5 = 2 (remainder 2) - 5 is not a factor
6. 6: 12 ÷ 6 = 2 (remainder 0) - 6 is a factor
7. 7: 12 ÷ 7 = 1 (remainder 5) - 7 is not a factor
8. 8: 12 ÷ 8 = 1 (remainder 4) - 8 is not a factor
9. 9: 12 ÷ 9 = 1 (remainder 3) - 9 is not a factor
10. 10: 12 ÷ 10 = 1 (remainder 2) - 10 is not a factor
11. 11: 12 ÷ 11 = 1 (remainder 1) - 11 is not a factor
12. 12: 12 ÷ 12 = 1 (remainder 0) - 12 is a factor

We stop at 12 because any number larger than the square root of 12 will have a corresponding factor we've already found. Therefore, the factors of 12 are 1, 2, 3, 4, 6, and 12.

Method 2: Prime Factorization

A more efficient method, especially for larger numbers, is prime factorization. This involves breaking down the number into its prime factors—numbers divisible only by 1 and themselves. Let's use 12 again:

1. Find a prime factor: The smallest prime factor of 12 is 2.
2. Divide: 12 ÷ 2 = 6
3. Repeat: The smallest prime factor of 6 is 2. 6 ÷ 2 = 3
4. Prime Factor: 3 is a prime number.

Therefore, the prime factorization of 12 is 2 x 2 x 3 (or 2² x 3).

Finding the Factors of 10

Now, let's apply this knowledge to find the factors of 10.

Using trial division:

1. 1: 10 ÷ 1 = 10 (remainder 0)
2. 2: 10 ÷ 2 = 5 (remainder 0)
3. 3: 10 ÷ 3 = 3 (remainder 1)
4. 4: 10 ÷ 4 = 2 (remainder 2)
5. 5: 10 ÷ 5 = 2 (remainder 0)
6. 6: 10 ÷ 6 = 1 (remainder 4)
7. 7: 10 ÷ 7 = 1 (remainder 3)
8. 8: 10 ÷ 8 = 1 (remainder 2)
9. 9: 10 ÷ 9 = 1 (remainder 1)
10. 10: 10 ÷ 10 = 1 (remainder 0)

The factors of 10 are 1, 2, 5, and 10.

Using prime factorization:

10 can be factored as 2 x 5. Both 2 and 5 are prime numbers. To find all factors, we consider all possible combinations of these prime factors:

• 2⁰ x 5⁰ = 1
• 2¹ x 5⁰ = 2
• 2⁰ x 5¹ = 5
• 2¹ x 5¹ = 10

Therefore, the factors of 10 are 1, 2, 5, and 10.

Answering the Question: Which Answer Choice Contains All the Factors of 10?

Without knowing the specific answer choices, we can definitively state that the correct answer choice must include the numbers 1, 2, 5, and 10. Any answer choice missing even one of these numbers is incorrect.

Expanding the Concept: Factors and Divisibility Rules

Understanding factors is essential for grasping other number theory concepts like divisibility rules. Divisibility rules provide quick ways to determine if a number is divisible by another without performing long division. For example:

• Divisibility by 2: A number is divisible by 2 if its last digit is even (0, 2, 4, 6, or 8).
• Divisibility by 3: A number is divisible by 3 if the sum of its digits is divisible by 3.
• Divisibility by 5: A number is divisible by 5 if its last digit is 0 or 5.
• Divisibility by 10: A number is divisible by 10 if its last digit is 0.

These rules are directly linked to factors. For instance, if a number is divisible by 2, then 2 is one of its factors.

Advanced Concepts: Greatest Common Factor (GCF) and Least Common Multiple (LCM)

The concept of factors extends to finding the Greatest Common Factor (GCF) and Least Common Multiple (LCM) of two or more numbers.

• GCF: The largest number that divides evenly into all the given numbers.
• LCM: The smallest number that is a multiple of all the given numbers.

Finding the GCF and LCM often involves prime factorization. Let's find the GCF and LCM of 12 and 18:

Prime Factorization:

• 12 = 2² x 3
• 18 = 2 x 3²

GCF: The common prime factors are 2 and 3. The lowest power of each common factor is 2¹ and 3¹. Therefore, the GCF(12, 18) = 2 x 3 = 6.

LCM: Consider all prime factors from both numbers (2 and 3). Use the highest power of each prime factor: 2² and 3². Therefore, the LCM(12, 18) = 2² x 3² = 4 x 9 = 36.

Conclusion: Mastering Factors for Mathematical Success

This in-depth exploration of factors demonstrates their fundamental role in number theory. Understanding how to identify factors, employing prime factorization, and applying this knowledge to concepts like GCF and LCM are crucial skills for anyone seeking a strong foundation in mathematics. Remember, the key to finding all the factors of a number lies in a systematic approach, whether through trial division or the more efficient prime factorization method. By mastering these techniques, you'll be well-equipped to confidently solve problems involving factors and unlock a deeper understanding of numbers. And, critically, you'll be able to correctly identify the answer choice that contains all the factors of 10, or any other number you encounter.