Prime factorization of 405.
Understand the Problem
The question is asking for the prime factorization of the number 405, which involves breaking it down into its prime number components. We will find which prime numbers multiply together to equal 405.
Answer
The prime factorization of 405 is $3^4 \times 5$.
Answer for screen readers
The prime factorization of 405 is $3^4 \times 5$.
Steps to Solve
- Start with the number 405
Begin by checking the smallest prime number, which is 2. Since 405 is odd, it cannot be divided by 2.
- Check the next prime number, which is 3
Check if 405 is divisible by 3. A quick way to do this is by adding the digits: $4 + 0 + 5 = 9$, and since 9 is divisible by 3, so is 405.
Now divide: $$ 405 \div 3 = 135 $$
- Factor 135 further
Now take 135 and check for divisibility by 3 again.
Add the digits: $1 + 3 + 5 = 9$, which is also divisible by 3.
Divide: $$ 135 \div 3 = 45 $$
- Continue factoring 45
Now take 45 and check for divisibility by 3 once more.
Add the digits: $4 + 5 = 9$, which is also divisible by 3.
Divide: $$ 45 \div 3 = 15 $$
- Factor 15
Next, take 15 and check for divisibility by 3 again.
Add the digits: $1 + 5 = 6$, which is divisible by 3.
Divide: $$ 15 \div 3 = 5 $$
- Final factorization
Now we have 5, which is a prime number.
Combine all the factors: $$ 405 = 3 \times 3 \times 3 \times 3 \times 5 $$ This can also be written using exponents: $$ 405 = 3^4 \times 5 $$
The prime factorization of 405 is $3^4 \times 5$.
More Information
The number 405 is interesting because it is not only a product of prime factors, but it is also a square composite number since $405 = 15^2 \times 3$ and appears in various bases in number theory.
Tips
- Failing to check for divisibility with smaller prime numbers properly.
- Forgetting to simplify or express factors using exponents.