What are the prime factors of 363?
Understand the Problem
The question is asking for the prime factors of the number 363. To solve this, we will perform a factorization process to determine which prime numbers multiply together to give 363.
Answer
The prime factorization of 363 is \( 3 \times 11^2 \).
Answer for screen readers
The prime factorization of 363 is ( 3 \times 11^2 ).
Steps to Solve
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Initial Division by Smallest Prime Start by dividing the number 363 by the smallest prime number, which is 2. Since 363 is odd, it is not divisible by 2.
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Next Prime Number (3) Check if 363 is divisible by 3. To do this, sum the digits: $3 + 6 + 3 = 12$. Since 12 is divisible by 3, 363 is also divisible by 3. Now perform the division: $$ 363 \div 3 = 121 $$
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Factorization of Result (121) Now we need to factor 121. Check divisibility by 2 and 3 again, which do not work.
Next, try 5 (not applicable here), then check 7 (it doesn't work), and then check 11: $$ 121 \div 11 = 11 $$
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Complete Factorization Now we have: $$ 121 = 11 \times 11 $$ Thus, our complete factorization of 363 becomes: $$ 363 = 3 \times 11 \times 11 $$
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Expressing in Prime Factorization Form Finally, write the prime factorization concisely as: $$ 363 = 3 \times 11^2 $$
The prime factorization of 363 is ( 3 \times 11^2 ).
More Information
The number 363 is an interesting case as it demonstrates how numbers can have multiple prime factors. Its prime factorization shows that 11 is squared, indicating that it appears twice in the factorization.
Tips
- A common mistake is to overlook checking divisibility by smaller prime numbers sequentially.
- Sometimes, the mistake is to forget that a prime number can repeat in the factorization, leading to incomplete factors.
