Finding the factors of a number is a fundamental concept in mathematics, and it’s essential to understand the process to solve various mathematical problems. In this case, we’ll explore how to find the factors of 294 using a simple method.
To begin with, let’s define what factors are. Factors are the numbers that divide a given number exactly without leaving a remainder. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12.
Now, let’s move on to finding the factors of 294. One way to do this is by using the prime factorization method. Prime factorization involves breaking down a number into its prime factors, which are the prime numbers that multiply together to give the original number.
Here’s how to find the prime factorization of 294:
- Start by dividing 294 by the smallest prime number, which is 2.
- 294 ÷ 2 = 147
- Since 147 is not divisible by 2, move on to the next prime number, which is 3.
- 147 ÷ 3 = 49
- Now, 49 is not divisible by 3, so move on to the next prime number, which is 7.
- 49 ÷ 7 = 7
- Now we have the prime factorization of 294: 2 × 3 × 7 × 7
Now that we have the prime factorization, we can find all the factors of 294 by multiplying the prime factors in different combinations. Here are all the factors of 294:
- 1 (since 1 is a factor of every number)
- 2 (prime factor)
- 3 (prime factor)
- 6 (2 × 3)
- 7 (prime factor)
- 14 (2 × 7)
- 21 (3 × 7)
- 42 (2 × 3 × 7)
- 49 (7 × 7)
- 98 (2 × 7 × 7)
- 147 (3 × 7 × 7)
- 294 (2 × 3 × 7 × 7)
And there you have it! The factors of 294 are 1, 2, 3, 6, 7, 14, 21, 42, 49, 98, 147, and 294.
To make it easier to understand, let’s use a simple table to list out all the factors:
| Factor | Description |
|---|---|
| 1 | Basic factor |
| 2 | Prime factor |
| 3 | Prime factor |
| 6 | 2 × 3 |
| 7 | Prime factor |
| 14 | 2 × 7 |
| 21 | 3 × 7 |
| 42 | 2 × 3 × 7 |
| 49 | 7 × 7 |
| 98 | 2 × 7 × 7 |
| 147 | 3 × 7 × 7 |
| 294 | 2 × 3 × 7 × 7 |
Now, let’s take a look at some examples to illustrate how to use the factors of 294 in real-world problems.
Example 1: Find the greatest common factor (GCF) of 294 and 168.
To find the GCF, list out the factors of both numbers and find the greatest common factor.
Factors of 294: 1, 2, 3, 6, 7, 14, 21, 42, 49, 98, 147, 294 Factors of 168: 1, 2, 3, 4, 6, 7, 8, 12, 14, 21, 24, 28, 42, 56, 84, 168
The greatest common factor of 294 and 168 is 42.
Example 2: Find the least common multiple (LCM) of 294 and 210.
To find the LCM, list out the multiples of both numbers and find the smallest common multiple.
Multiples of 294: 294, 588, 882, 1176,… Multiples of 210: 210, 420, 630, 840,…
The least common multiple of 294 and 210 is 8820.
In conclusion, finding the factors of 294 is a straightforward process that involves prime factorization and multiplying the prime factors in different combinations. By understanding how to find the factors of a number, you can solve various mathematical problems, including finding the greatest common factor and least common multiple of two numbers.
What is the prime factorization of 294?
+The prime factorization of 294 is 2 × 3 × 7 × 7.
How many factors does 294 have?
+294 has 12 factors: 1, 2, 3, 6, 7, 14, 21, 42, 49, 98, 147, and 294.
What is the greatest common factor of 294 and 168?
+The greatest common factor of 294 and 168 is 42.
By following these steps and examples, you can easily find the factors of 294 and apply this knowledge to solve various mathematical problems. Remember to practice regularly to reinforce your understanding of this concept.
