Problem 12 - Prealgebra

Question

Factor each of the following into a product of prime factors. $$210$$

Step-by-Step Solution

Answer

210 = 2 × 3 × 5 × 7

Step 1: Identify a Pair of Factors

Start by finding two numbers that multiply together to get the original number, 210. Let’s take 2 and 105, since 210 is even and divisible by 2.

Step 2: Break Down Non-Prime Factor

Check if 105 can be further broken down into factors. Recognize that 105 is divisible by 3 because adding its digits (1 + 0 + 5) equals 6, which is divisible by 3. So, 105 can be divided into 3 and 35.

Step 3: Continue Factoring Non-Prime

Now factor 35, which can be divided into 5 and 7, both of which are prime numbers.

Step 4: Collect All Prime Factors

You have now obtained all prime factors of 210: 2, 3, 5, and 7. Therefore, 210 can be expressed as a product of these prime factors: \[ 210 = 2 imes 3 imes 5 imes 7 \]

Key Concepts

Understanding Factors

In mathematics, finding factors means breaking down a number into smaller numbers that, when multiplied together, give back the original number. For example, if we take the number 210, the factors are pairs of numbers that multiply together to give us 210. When looking for factors, it's helpful to start by determining if smaller numbers divide evenly into the original number. Factors are integral to simplifying math problems, particularly when prime factorizing numbers. Here’s a simple way to interpret the process:

• Begin with the smallest prime number (2) and check for divisibility.
• Continue testing divisibility with other numbers, moving up sequentially.
• Use each factor to divide the number, gradually reducing its size to simpler terms.

Breaking down a number into its factors can greatly simplify mathematical operations, especially in prime factorization.

What are Prime Numbers?

Prime numbers are a fundamental concept in mathematics. They are numbers greater than 1, which can only be divided by 1 and themselves without leaving a remainder. This means they have exactly two distinct positive divisors. For instance, numbers like 2, 3, 5, and 7 are prime numbers. The unique nature of prime numbers makes them the building blocks for all numbers, essentially the 'atoms' in the world of numbers. Recognizing prime numbers is crucial in the process of prime factorization because every number can be expressed as a product of primes. Here are key characteristics of prime numbers:

• They cannot be formed by multiplying two smaller natural numbers.
• Except for 2, all prime numbers are odd, as even numbers will have at least three divisors (1, 2, and themselves).
• Knowing the list of smaller prime numbers helps in prime factorization.

Understanding prime numbers gives insight into the foundational makeup of all numbers, as demonstrated when factorizing numbers like 210 into its primes: 2, 3, 5, and 7.

Applying Divisibility Rules

Divisibility rules are shortcuts that help determine if one number can be divided by another without leaving a remainder. These rules are incredibly useful when factorizing numbers, as they quickly reveal which numbers to test as factors. Here are some useful divisibility rules:

• Divisibility by 2: A number is divisible by 2 if it is even.
• Divisibility by 3: A number is divisible by 3 if the sum of its digits is divisible by 3.
• Divisibility by 5: A number is divisible by 5 if it ends in 0 or 5.

In the example of 210, we used these rules to find its factors: - Since 210 is even, it’s divisible by 2. - The sum of the digits in 105 (1 + 0 + 5) is 6, which is divisible by 3, hinting division by 3. - Finally, the last digit of 210 is a 0, confirming its divisibility by 5. Mastering divisibility rules simplifies the factorization process and expedites finding a number’s prime factors.