What is the prime factorization of 294?
Understand the Problem
The question is asking for the prime factorization of the number 294, which means finding the prime numbers that multiply together to give the original number.
Answer
The prime factorization of 294 is $2 \times 3 \times 7^2$.
Answer for screen readers
The prime factorization of 294 is $2 \times 3 \times 7^2$.
Steps to Solve
- Start with the given number
We begin with the number 294. We need to determine if it's divisible by the smallest prime number, which is 2.
- Divide by the smallest prime
Since 294 is an even number, we can divide it by 2:
$$ 294 \div 2 = 147 $$
So, we have:
$$ 294 = 2 \times 147 $$
- Factor the next number
Now we take the quotient, 147, and check for its prime factors. The sum of digits in 147 is 12, which is divisible by 3. Thus, we can divide by 3:
$$ 147 \div 3 = 49 $$
So now we have:
$$ 294 = 2 \times 3 \times 49 $$
- Factor the remaining number
Next, we need to factor 49. We recognize that 49 is a perfect square:
$$ 49 = 7 \times 7 $$
So, we can now write:
$$ 294 = 2 \times 3 \times 7 \times 7 $$
- Express with exponents
We can simplify our answer by expressing the repeated factor with an exponent. Thus, the prime factorization becomes:
$$ 294 = 2 \times 3 \times 7^2 $$
The prime factorization of 294 is $2 \times 3 \times 7^2$.
More Information
The prime factorization helps in various fields such as cryptography, number theory, and finding the greatest common divisors (GCD). It shows how a number can be broken down into building blocks, which are the prime numbers.
Tips
- Forgetting to check divisibility by small prime numbers, such as 2 or 3.
- Not recognizing that some numbers can be factored further (e.g. realizing 49 is $7^2$).
- Confusing composite numbers with prime numbers.
