Prime factorization of 363
Understand the Problem
The question is asking for the prime factorization of the number 363, which involves breaking it down into its prime factors.
Answer
$3 \times 11^2$
Answer for screen readers
The prime factorization of 363 is $3 \times 11^2$.
Steps to Solve
-
Start with the number
Begin with the number 363. -
Check for divisibility by the smallest prime number (2)
The number 363 is odd, so it's not divisible by 2. -
Check for divisibility by the next prime number (3)
To check if 363 is divisible by 3, we can add the digits:
$$ 3 + 6 + 3 = 12 $$
Since 12 is divisible by 3, we can divide:
$$ 363 \div 3 = 121 $$ -
Factor the quotient (121)
Now we need to factor 121. We can check for divisibility by the next prime numbers. -
Check divisibility by 2 and 3
121 is not even, so it is not divisible by 2. The sum of the digits $1 + 2 + 1 = 4$ is not divisible by 3, so it's also not divisible by 3. -
Check divisibility by 5
121 does not end in 0 or 5, so it is not divisible by 5. -
Check divisibility by 7
$$ 121 \div 7 \approx 17.2857 $$
This is not a whole number, thus 121 is not divisible by 7. -
Check divisibility by 11
$$ 121 \div 11 = 11 $$
Since 11 is prime, we can write:
$$ 121 = 11 \times 11 $$ -
Combine the factors
Now, we can combine the factors from the divisions: $$ 363 = 3 \times 11 \times 11 $$ -
Write in exponential form
The final prime factorization of 363 can also be expressed as: $$ 363 = 3 \times 11^2 $$
The prime factorization of 363 is $3 \times 11^2$.
More Information
The prime factorization of a number helps identify its building blocks, which are the prime numbers that multiply to give that number. It's essential in many areas of mathematics, including number theory and algebra.
Tips
- Failing to check for divisibility by progressively larger prime numbers can lead to not finding the correct prime factorization. It's important to check all primes systematically.
- Miscalculating when dividing can also lead to errors in finding factors.
