What is the prime factorization of 165?

Understand the Problem

The question is asking for the prime factors of the number 165. This involves breaking down the number into its prime components, which are the prime numbers that multiply together to give the original number.

Answer

The prime factors of 165 are $3$, $5$, and $11$.

Steps to Solve

Start with the smallest prime number
Check if 2 is a factor of 165. Since 165 is odd, it is not divisible by 2.
Check the next prime number
Now check if 3 is a factor of 165. To do this, add the digits of 165: $1 + 6 + 5 = 12$. Since 12 is divisible by 3, this means 165 is also divisible by 3.
Divide by 3
Perform the division:
$$ \frac{165}{3} = 55 $$
So, 3 is one of the prime factors.
Factor the result (55)
Now we need to find the prime factors of 55. Start with the smallest prime number again, which is 2. 55 is odd, so it is not divisible by 2.
Check the next prime number (3)
Check if 3 is a factor of 55 by adding the digits: $5 + 5 = 10$. Since 10 is not divisible by 3, 55 is not divisible by 3.
Check the next prime number (5)
Check if 5 is a factor of 55. The last digit of 55 is 5, which means it is divisible by 5.
Divide by 5
Perform the division:
$$ \frac{55}{5} = 11 $$
So, 5 is another prime factor.
Check the result (11)
Finally, we need to check if 11 is a prime number. The only divisors of 11 are 1 and 11 itself, making it a prime number.
Combine the prime factors
The prime factors of 165 are 3, 5, and 11.

The prime factors of 165 are $3$, $5$, and $11$.

More Information

Prime factorization is a fundamental concept in number theory. The process illustrates how composite numbers can be expressed as products of prime numbers, which is useful in various areas of mathematics, like simplifying fractions and finding greatest common divisors.

Tips

Forgetting to check for divisibility by all prime numbers.
Stopping the factorization process before confirming that all resulting factors are primes.

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