Problem 22 - Fundamentals of Mathematics

Question

Find the greatest common factor (GCF) of the numbers. \(147,343,\) and 231

Step-by-Step Solution

Answer

The GCF of 147, 343, and 231 is 7.

Step 1: Prime Factorization of 147

To begin, we need to find the prime factorization of 147. Start by dividing 147 by the smallest prime number, 3: \[147 \div 3 = 49\] Next, factor 49, which is a perfect square and can be factored as: \[49 = 7 \times 7\] So, the prime factorization of 147 is: \[147 = 3 \times 7^2\]

Step 2: Prime Factorization of 343

Now, let's find the prime factorization of 343. Recognize that 343 is a power of 7: \[343 = 7 \times 7 \times 7 = 7^3\] Thus, the prime factorization of 343 is: \[343 = 7^3\]

Step 3: Prime Factorization of 231

Proceed with the prime factorization of 231. Start by dividing by 3: \[231 \div 3 = 77\] Further factor 77 by dividing it by the prime number 7: \[77 \div 7 = 11\] Finally, 11 is a prime number. So, the prime factorization of 231 is: \[231 = 3 \times 7 \times 11\]

Step 4: Identify Common Prime Factors

Using the prime factorizations from previous steps: - 147: \(3 \times 7^2\) - 343: \(7^3\) - 231: \(3 \times 7 \times 11\) The common prime factor among all three numbers is 7.

Step 5: Find the Greatest Common Factor

Since the common prime factor is 7 and it occurs at the lowest power of 1 in all factorizations, the greatest common factor (GCF) is: \[\text{GCF} = 7^1 = 7\]

Key Concepts

Prime Factorization

Prime factorization is the process of breaking down a composite number into its prime numbers. A prime number is a number greater than one, which has no divisors other than 1 and itself. To perform prime factorization, you divide the number by the smallest prime, then continue dividing by prime numbers until the quotient is a prime number. This gives you the product of prime numbers that multiply to the original number. For example, in the exercise provided, the prime factorization of 147 involves dividing it by 3 to get 49, and then breaking down 49 into 7 and 7, resulting in:

• 147 as a product of primes: \(147 = 3 \times 7^2\)
• 343 can be expressed as \(343 = 7^3\) since 343 equals 7 cubed.
• 231 is factored by dividing by 3, then 7, finally leaving 11, giving \(231 = 3 \times 7 \times 11\).

Notice how breaking down numbers like these makes finding the greatest common factor easier.

Common Prime Factors

The concept of common prime factors revolves around identifying the prime numbers that appear in all the factorizations. It's like finding a common thread woven through different numbers. To spot these common factors, you examine the list of primes from each number’s factorization. In the case of 147, 343, and 231:

• 147 consists of the primes \(3\) and \(7^2\).
• 343 is simply \(7^3\).
• 231 contains the primes \(3\), \(7\), and \(11\).

The common prime number here is 7, which is present in the prime factorizations of all three numbers. Remember, to find the greatest common factor, the exponents of these common primes in each number need to be compared, and the lowest exponent taken. Here, the lowest power of 7 found is \(7^1\), simplifying the GCF to 7.

Divisibility

Divisibility is a foundational concept in arithmetic, which tells us if one number can be divided by another without a remainder. In prime factorization, recognizing a number's divisibility by smaller primes is essential. For instance:

• When we say 147 is divisible by 3, we mean that dividing it gives a whole number, 49, without leftover.
• Similarly, 343 is easily divisible by 7 since it's a power of 7.
• 231 is divisible by both 3 and 7, breaking down further into 7 x 11 as well.

Understanding divisibility helps in quickly identifying factors and determining relationships between different numbers. This understanding paves the way to finding common factors directly: any number that leaves no remainder when divided by a certain integer indicates a level of divisibility by that integer.