Problem 22 - Elementary and Intermediate Algebra

Question

Find the prime factorization of each number. If the number is prime, state this. $$ 15 $$

Step-by-Step Solution

Answer

The prime factorization of 15 is 3 and 5.

Step 1: Understand Prime Factorization

Prime factorization involves breaking down a number into its prime factors, which are prime numbers that multiply together to give the original number.

Step 2: Find the Smallest Prime Number

Start with the smallest prime number, which is 2, and check if it divides 15 without leaving a remainder. Since 15 is not divisible by 2, move to the next smallest prime number, which is 3.

Step 3: Divide by the Prime Number

Check if 15 is divisible by 3. Since 15 divided by 3 equals 5, which is a whole number, 3 is one of the prime factors.

Step 4: Check the Quotient

The quotient from the previous step is 5. Check if 5 is a prime number. Since 5 cannot be divided evenly by any number other than 1 and itself, it is a prime number.

Step 5: Write the Prime Factorization

Combine the prime factors found. Therefore, the prime factorization of 15 is 3 and 5.

Key Concepts

Prime Numbers

Prime numbers are special numbers greater than 1 that have no divisors other than 1 and themselves.
In other words, they can't be formed by multiplying two smaller natural numbers.

Division

Division is the process of determining how many times one number is contained within another.
It is fundamental to breaking down numbers into smaller components, especially in prime factorization.

Factors

Factors are numbers that you can multiply together to get another number.
Prime factors are specifically prime numbers that multiply together to get the original number.