To find the factors of 294, we first need to understand what factors are. Factors are the numbers that can be multiplied together to get a specific number. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12 because these numbers can be multiplied in pairs to give 12.
Let’s break down 294 into its prime factors to easily identify all its factors. The prime factorization of a number involves breaking it down into a product of prime numbers.
Step 1: Prime Factorization of 294
To start, we look for the smallest prime number that can divide 294 without leaving a remainder. We test the first few prime numbers: 2, 3, 5, 7, 11, etc., until we find one that works.
- 2 is the smallest prime number, and 294 is even, so it’s divisible by 2: 294 ÷ 2 = 147.
- Now, we look at 147. It’s odd, so 2 won’t work. The next prime number is 3, and 147 is divisible by 3: 147 ÷ 3 = 49.
- 49 is not divisible by 2 or 3. The next prime number is 7, and 49 is divisible by 7: 49 ÷ 7 = 7.
- 7 is a prime number itself.
So, the prime factorization of 294 is 2 × 3 × 7 × 7, or more succinctly, 2 × 3 × 7^2.
Step 2: Finding All Factors
Now that we have the prime factorization (2 × 3 × 7^2), we can find all the factors of 294 by combining these prime factors in all possible ways:
- 1 (since 1 is a factor of every number)
- 2
- 3
- 7
- 2 × 3 = 6
- 2 × 7 = 14
- 3 × 7 = 21
- 2 × 3 × 7 = 42
- 7 × 7 = 49 (since 7^2 is part of the prime factorization)
- 2 × 7 × 7 = 98
- 3 × 7 × 7 = 147
- 2 × 3 × 7 × 7 = 294 (the number itself)
Conclusion
The factors of 294 are 1, 2, 3, 6, 7, 14, 21, 42, 49, 98, 147, and 294. By following the prime factorization method, we can easily find all the factors of any given number, making it a powerful tool for understanding the building blocks of numbers.
