Problem 11 - Fundamentals of Mathematics

Question

Find the greatest common factor (GCF) of the numbers. 35 and 175

Step-by-Step Solution

Answer

The GCF of 35 and 175 is 35.

Step 1: List the factors of each number

First, list out all the factors of each number. For 35, the factors are 1, 5, 7, and 35. For 175, the factors are 1, 5, 7, 25, 35, and 175.

Step 2: Identify the common factors

Look at the lists of factors from each number and identify which factors are common to both. The common factors of 35 and 175 are 1, 5, and 35.

Step 3: Determine the greatest common factor

From the list of common factors, determine which is the greatest. The greatest common factor of 35 and 175 is 35.

Key Concepts

Factors

In mathematics, a factor is a whole number that can be multiplied by another whole number to produce a given number. For example, if we take the number 35, factors are the numbers that can divide 35 evenly with no remainder. These are:

• 1
• 5
• 7
• 35

To find the factors, we can start by dividing the number with smaller numbers (starting from 1) to see if the quotient is a whole number.
Remember, every number is divisible by 1 and itself, so these are always factors.
This process helps us identify all possible pairs of numbers that can be multiplied to obtain the original number.
Understanding factors is crucial because they lay the groundwork for finding common factors and eventually the greatest common divisor (GCD) of multiple numbers.

Common Factors

A common factor is any factor that two or more numbers share. To find common factors, you list the factors of each number and look for numbers that appear in each list.
For instance, with our numbers 35 and 175:

• Factors of 35: 1, 5, 7, 35
• Factors of 175: 1, 5, 7, 25, 35, 175

After listing, we identify the shared factors:

• 1
• 5
• 35

Common factors are important because they help us understand how numbers relate to each other through division.
They also indicate how two numbers can be evenly grouped or partitioned. Recognizing these common factors is a key step before finding the greatest one.

Greatest Common Divisor

The greatest common divisor (GCD), also known as the greatest common factor (GCF), is the largest number that can divide two or more numbers without leaving a remainder.
To find the GCD, we rely on the list of common factors.
For numbers 35 and 175, the common factors are 1, 5, and 35.
Among these, the largest number is 35, which is the GCD.
Finding the GCD is particularly useful in simplifying fractions, as it allows us to reduce them to their simplest form.
It also helps in solving problems that involve divisibility and can be handy in real-world situations like distributing items evenly. When two numbers share a GCD, they can be broken down into their simplest shared components efficiently.