Problem 12 - Intermediate Algebra

Question

In the following exercises, find the prime factorization. 627

Step-by-Step Solution

Answer

3 × 3 × 7 × 29

Step 1: Identify if the number is divisible by the smallest prime, 2

Since 627 is odd, it is not divisible by 2. Move to the next smallest prime.

Step 2: Check divisibility by 3

To check if 627 is divisible by 3, add the digits of the number: 6 + 2 + 7 = 15. Since 15 is divisible by 3, 627 is also divisible by 3. Proceed to divide 627 by 3: 627 ÷ 3 = 209

Step 3: Check divisibility of 209 by smaller primes starting from 2

209 is odd, so it's not divisible by 2. Adding the digits of 209 (2 + 0 + 9 = 11) shows it's not divisible by 3 either. It doesn’t end in 0 or 5, so it is not divisible by 5.

Step 4: Check divisibility by 7

To check if 209 is divisible by 7, we perform the division: 209 ÷ 7 = 29. Since 29 is an integer, it confirms that 209 is divisible by 7. The result of the division gives us 29.

Step 5: Verify that 29 is a prime number

29 is a prime number since it has no divisors other than 1 and itself. So the prime factorization is complete.

Step 6: Write the full prime factorization

Combine all the prime factors obtained in the previous steps. The prime factorization of 627 is: 3 × 3 × 7 × 29

Key Concepts

prime numbers

Prime numbers are the building blocks of all natural numbers. A prime number is a number that has exactly two distinct, positive divisors: 1 and itself.
This means a prime number cannot be divided evenly by any other number except these two. Some examples are 2, 3, 5, 7, 11, and 29.
When you factorize a number into primes, you break it down into the prime numbers that multiply together to make the original number.
For example, in the factorization of 627: when we reach 29, we stop because 29 is a prime number.

divisibility rules

Divisibility rules can help you determine if a number is divisible by another without performing full division. These rules simplify the process.
Writing out all possible factors can be daunting, but divisibility rules come in handy. Let's consider a few:

• Rule for 2: A number is divisible by 2 if it is even (ends with 0, 2, 4, 6, or 8).
• Rule for 3: A number is divisible by 3 if the sum of its digits is divisible by 3.
• Rule for 5: A number is divisible by 5 if it ends with 0 or 5.

In the solution to 627, the rule for 3 helped us break it down easily by summing the digits (6+2+7=15). Since 15 is divisible by 3, so is 627.

factorization

Factorization is the process of breaking down a number into its constituent factors. When those factors are all prime numbers, we call it prime factorization.
The goal is to write the number as a product of prime numbers. For example, finding the prime factorization of 627 involves:

• Checking divisibility by small prime numbers.
• Dividing step by step until you are left with a prime number.

This step-by-step approach ensures you don't miss any factors, and the final product is the complete factorization.
For 627, the factors are 3, 3, 7, and 29, because 3 x 3 x 7 x 29 represents the original number when multiplied together.

basic arithmetic

Basic arithmetic involves operations like addition, subtraction, multiplication, and division. Mastery of these is essential for tasks like prime factorization.
For example:

• Addition: Used here to check divisibility by 3 (summing the digits of 627 as 6 + 2 + 7 = 15).
• Multiplication: Used to combine the prime factors (3 x 3 x 7 x 29 = 627).
• Division: Crucial for breaking down the number step by step (627 ÷ 3 = 209, then 209 ÷ 7 = 29).

Each operation plays a role in finding the prime factors, demonstrating the importance of understanding basic arithmetic in the process.