what are the factors of 12
What are the factors of 12?
Answer: Factors of a number are all the whole numbers that can be multiplied in pairs to produce that original number. In the case of 12, there are several pairs of numbers that can be multiplied to result in 12. The factors of 12 are 1, 2, 3, 4, 6, and 12. To understand this better, let’s break it down step by step.
Understanding Factors
Definition: A factor of a number is an integer that divides the number without leaving a remainder. For example, in simpler terms, if you can divide one number by another and the division yields a whole number without leftovers, then both numbers are factors of the original one.
Step-by-step Calculation of Factors of 12
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Identifying Pairwise Factors: Start by examining the smallest number, which is 1, and the largest number of any factor list, which is the number itself, in this case, 12.
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Calculation Process:
- 1 \times 12 = 12: This means both 1 and 12 are factors.
- 2 \times 6 = 12: This indicates that 2 and 6 are also factors.
- 3 \times 4 = 12: Thus, 3 and 4 are factors too.
By examining each of these pairs, we ascertain that the complete list of factors includes 1, 2, 3, 4, 6, and 12.
Verification Method
To ensure that you’ve correctly identified all factors, test each number between 1 and 12:
- For 1 through 12:
- If you divide 12 by a number and get a quotient with no remainder, then that number is a factor.
- Examples:
- 12 \div 1 = 12, remainder 0, so 1 is a factor.
- 12 \div 2 = 6, remainder 0, so 2 is a factor.
- 12 \div 3 = 4, remainder 0, so 3 is a factor.
- 12 \div 4 = 3, remainder 0, so 4 is a factor.
- 12 \div 6 = 2, remainder 0, so 6 is a factor.
- 12 \div 12 = 1, remainder 0, so 12 is a factor.
Prime Factorization Approach
A deeper understanding of factors can be obtained through prime factorization (expressing a number as a product of its prime numbers):
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Prime Factorization of 12:
- Start by dividing by the smallest prime: 12 ÷ 2 = 6.
- Continue dividing by 2: 6 ÷ 2 = 3.
- Finally, 3 is a prime number.
Combining these, you get 12 = 2^2 \times 3. The prime factors are 2 and 3.
Real-life Analogy
Consider you have 12 apples and want to divide them into groups without cutting any apples. The possible group sizes where each group can have the exact number of apples based entirely on whole numbers are 1, 2, 3, 4, 6, or 12 - meaning these sizes are the factors of 12.
Factors in a Table for Clarity
| Number Pair | Product |
|---|---|
| 1 and 12 | 12 |
| 2 and 6 | 12 |
| 3 and 4 | 12 |
The table clearly shows how each combination of factors multiplies to 12.
Interactive Consideration
Ask yourself, “Can this pair divide 12 evenly?” This question helps cement the concept of quantitatively verifying factors, ensuring comprehension.
Motivation and Encouragement
Remember, understanding factors is a fundamental step in more advanced mathematical concepts such as prime numbers, greatest common divisors, and least common multiples. By mastering these basics, you pave the way for future math success. Keep practicing with different numbers to build confidence!
In summary, the factors of 12 are 1, 2, 3, 4, 6, and 12, and understanding these involves examining pairwise multiplication and utilizing division to verify. Prime factorization further breaks down numbers into prime roots, expanding your toolkit for tackling math problems. Keep exploring, and you’ll enhance your math skills progressively!