What is the prime factorization of 336?
Understand the Problem
The question is asking for the prime factorization of the number 336, meaning we need to express 336 as a product of its prime factors.
Answer
The prime factorization of 336 is \(2^4 \times 3^1 \times 7^1\).
Answer for screen readers
The prime factorization of 336 is (2^4 \times 3^1 \times 7^1).
Steps to Solve
- Start with the number 336
To find the prime factorization of 336, we begin by dividing it by the smallest prime number, which is 2.
- Divide by 2
Since 336 is even, we can divide it by 2:
$$ 336 \div 2 = 168 $$
Now, we record 2 as one of the prime factors.
- Continue dividing by 2
Next, we divide 168 by 2 again:
$$ 168 \div 2 = 84 $$
We record another 2.
- Keep dividing by 2
Now, we divide 84 by 2:
$$ 84 \div 2 = 42 $$
Again, we add another 2 to our list of prime factors.
- One more division by 2
We now divide 42 by 2:
$$ 42 \div 2 = 21 $$
We have another 2, and now 21 is not divisible by 2.
- Divide by the next smallest prime, 3
Next, we try dividing 21 by the next smallest prime number, which is 3:
$$ 21 \div 3 = 7 $$
We now record 3 as another prime factor.
- End with the last prime number
Finally, 7 is a prime number itself, so we add it to our list of prime factors.
- Compile the prime factors
Now, we can write the complete factorization of 336:
$$ 336 = 2^4 \times 3^1 \times 7^1 $$
The prime factorization of 336 is (2^4 \times 3^1 \times 7^1).
More Information
Prime factorization helps in understanding the composition of numbers and is useful in topics such as least common multiples, greatest common divisors, and number theory. The number 336 can be represented in many ways, but the prime factorization is a unique product of primes.
Tips
- Forgetting to continue dividing by the smallest prime after each step.
- Not recognizing a number is prime can lead to incorrect factors being included.
- Miscounting the number of times a prime number divides the original number.
