Problem 16 - Fundamentals of Mathematics

Question

Find the greatest common factor (GCF) of the numbers. 99 and 135

Step-by-Step Solution

Answer

The GCF of 99 and 135 is 9.

Step 1: Prime Factorization of 99

To find the prime factors of 99, we divide the number by the smallest prime numbers. - Start with 2: 99 is odd, so it is not divisible by 2.- Next, divide by 3: 99 ÷ 3 = 33, so 3 is a factor.- Divide 33 by 3 again: 33 ÷ 3 = 11, so another 3 is a factor.- Finally, 11 is a prime number.Thus, the prime factors of 99 are: \[ 99 = 3^2 \times 11 \]

Step 2: Prime Factorization of 135

Now, let's find the prime factors of 135. - Start with 2: 135 is odd, not divisible by 2.- Divide by 3: 135 ÷ 3 = 45, so 3 is a factor.- Divide 45 by 3 again: 45 ÷ 3 = 15.- Divide 15 by 3 again: 15 ÷ 3 = 5.- Finally, 5 is a prime number.Thus, the prime factors of 135 are: \[ 135 = 3^3 \times 5 \]

Step 3: Identify Common Factors

Now check the prime factorizations to find common factors:- 99 has the factors \(3^2 \times 11\).- 135 has the factors \(3^3 \times 5\).The common factor between these two numbers is 3. Since both have at least two 3's, the highest power of 3 that is common is \(3^2\).

Step 4: Calculate the GCF

The greatest common factor (GCF) is the product of all common factors raised to their lowest powers: The only common factor is 3 raised to the power of 2.So the GCF is:\[ 3^2 = 9 \]

Key Concepts

Understanding Prime Factorization

Prime factorization is a method used to express a number as a product of its prime numbers. Prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves. To find the prime factors of a number, start by dividing by the smallest prime, which is 2. If the number is not divisible by 2, like our numbers 99 and 135, move to the next smallest prime, which is 3.
99 and 135 are both not divisible by 2. But when we test them with 3:

• 99 can be divided by 3 to get 33, and further division by 3 results in 11. Thus, the prime factors are 3, 3, and 11.
• 135 is divided by 3 to get 45, then 15, and finally 5, leaving us with the prime factors 3, 3, 3, and 5.

It's like breaking a number down to the building blocks that comprise it, solely using primes.

Identifying Common Factors

Once both numbers are broken down into their prime factors, the next step is to identify the common factors.
For numbers like 99 and 135:

• The prime factorization of 99 is expressed as \(3^2 \times 11\).
• The prime factorization of 135 is expressed as \(3^3 \times 5\).

The common factor in both factorizations is the repeated prime number 3. The least repeated amount is crucial here because the GCF is determined by the lowest power of these common factors. In our case, both have common factor 3, but 99 has it twice. Thus, the greatest power they share is \(3^2\).

Mastering the Math of Problem-Solving

When tackling math problems, especially those involving finding the greatest common factor (GCF), breaking the problem into manageable steps can simplify the task. 1. **Prime Factorization**: Begin by decomposing each number into its prime factors. 2. **Find Common Factors**: Compare these factors to find what they have in common. 3. **Calculate the GCF**: Use the common factors raised to their lowest powers to compute the GCF. This systematic approach can be applied to a variety of math problems beyond just GCF. By practicing such steps consistently:

• You develop a strong foundation in mathematical problem-solving.
• You build confidence to approach increasingly complex problems.
• You gain skills useful for both academic pursuits and everyday life challenges.

In our example with 99 and 135, this structured approach revealed the GCF as 9, making it a more intuitive and straightforward task.